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Changeset 11543 for NEMO/trunk – NEMO

Changeset 11543 for NEMO/trunk


Ignore:
Timestamp:
2019-09-13T15:57:52+02:00 (5 years ago)
Author:
nicolasmartin
Message:

Implementation of convention for labelling references + files renaming
Now each reference is supposed to have the information of the chapter in its name
to identify quickly which file contains the reference (\label{$prefix:$chap_...)

Rename the appendices from 'annex_' to 'apdx_' to conform with the prefix used in labels (apdx:...)
Suppress the letter numbering

Location:
NEMO/trunk/doc/latex
Files:
19 edited
7 moved

Legend:

Unmodified
Added
Removed
  • NEMO/trunk/doc/latex/NEMO/main/appendices.tex

    r11330 r11543  
    11 
    2 \subfile{../subfiles/annex_A}             %% Generalised vertical coordinate 
    3 \subfile{../subfiles/annex_B}             %% Diffusive operator 
    4 \subfile{../subfiles/annex_C}             %% Discrete invariants of the eqs. 
    5 \subfile{../subfiles/annex_iso}            %% Isoneutral diffusion using triads 
    6 \subfile{../subfiles/annex_DOMAINcfg}     %% Brief notes on DOMAINcfg 
     2\subfile{../subfiles/apdx_s_coord}      %% Generalised vertical coordinate 
     3\subfile{../subfiles/apdx_diff_opers}   %% Diffusive operators 
     4\subfile{../subfiles/apdx_invariants}   %% Discrete invariants of the eqs. 
     5\subfile{../subfiles/apdx_triads}       %% Isoneutral diffusion using triads 
     6\subfile{../subfiles/apdx_DOMAINcfg}    %% Brief notes on DOMAINcfg 
    77 
    88%% Not included 
     
    1010%\subfile{../subfiles/chap_DIU} 
    1111%\subfile{../subfiles/chap_conservation} 
    12 %\subfile{../subfiles/annex_E}            %% Notes on some on going staff 
    13  
     12%\subfile{../subfiles/apdx_algos}   %% Notes on some on going staff 
  • NEMO/trunk/doc/latex/NEMO/main/chapters.tex

    r11522 r11543  
    1313\subfile{../subfiles/chap_STO}            %% Stochastic param. 
    1414\subfile{../subfiles/chap_misc}           %% Miscellaneous topics 
    15 \subfile{../subfiles/chap_CONFIG}         %% Predefined configurations 
     15\subfile{../subfiles/chap_cfgs}           %% Predefined configurations 
    1616 
    1717%% Not included 
  • NEMO/trunk/doc/latex/NEMO/main/introduction.tex

    r11522 r11543  
    5656 
    5757\begin{description} 
    58 \item [\nameref{chap:PE}] presents the equations and their assumptions, the vertical coordinates used, 
     58\item [\nameref{chap:MB}] presents the equations and their assumptions, the vertical coordinates used, 
    5959and the subgrid scale physics. 
    6060The equations are written in a curvilinear coordinate system, with a choice of vertical coordinates 
     
    6363Dimensional units in the meter, kilogram, second (MKS) international system are used throughout. 
    6464The following chapters deal with the discrete equations. 
    65 \item [\nameref{chap:STP}] presents the model time stepping environment. 
     65\item [\nameref{chap:TD}] presents the model time stepping environment. 
    6666it is a three level scheme in which the tendency terms of the equations are evaluated either 
    6767centered in time, or forward, or backward depending of the nature of the term. 
     
    123123\item [\nameref{chap:ASM}] describes how increments produced by 
    124124data \textbf{A}s\textbf{S}i\textbf{M}ilation may be applied to the model equations. 
     125\item [\nameref{chap:STO}] 
    125126\item [\nameref{chap:MISC}] (including solvers) 
    126 \item [\nameref{chap:CFG}] provides finally a brief introduction to 
     127\item [\nameref{chap:CFGS}] provides finally a brief introduction to 
    127128the pre-defined model configurations 
    128129(water column model \texttt{C1D}, ORCA and GYRE families of configurations). 
     
    133134 
    134135\begin{description} 
    135 \item [\nameref{apdx:s_coord}] 
    136 \item [\nameref{apdx:diff_oper}] 
    137 \item [\nameref{apdx:invariants}] 
    138 \item [\nameref{apdx:triads}] 
    139 \item [\nameref{apdx:DOMAINcfg}] 
    140 \item [\nameref{apdx:coding}] 
     136\item [\nameref{apdx:SCOORD}] 
     137\item [\nameref{apdx:DIFFOPERS}] 
     138\item [\nameref{apdx:INVARIANTS}] 
     139\item [\nameref{apdx:TRIADS}] 
     140\item [\nameref{apdx:DOMCFG}] 
     141\item [\nameref{apdx:CODING}] 
    141142\end{description} 
  • NEMO/trunk/doc/latex/NEMO/subfiles/apdx_DOMAINcfg.tex

    r11529 r11543  
    66% ================================================================ 
    77\chapter{A brief guide to the DOMAINcfg tool} 
    8 \label{apdx:DOMAINcfg} 
     8\label{apdx:DOMCFG} 
    99 
    1010\chaptertoc 
     
    121121The reference coordinate transformation $z_0(k)$ defines the arrays $gdept_0$ and 
    122122$gdepw_0$ for $t$- and $w$-points, respectively. See \autoref{sec:DOMCFG_sco} for the 
    123 S-coordinate options.  As indicated on \autoref{fig:index_vert} \jp{jpk} is the number of 
     123S-coordinate options.  As indicated on \autoref{fig:DOM_index_vert} \jp{jpk} is the number of 
    124124$w$-levels.  $gdepw_0(1)$ is the ocean surface.  There are at most \jp{jpk}-1 $t$-points 
    125125inside the ocean, the additional $t$-point at $jk = jpk$ is below the sea floor and is not 
     
    421421The depth field $h$ is not necessary the ocean depth, 
    422422since a mixed step-like and bottom-following representation of the topography can be used 
    423 (\autoref{fig:z_zps_s_sps}) or an envelop bathymetry can be defined (\autoref{fig:z_zps_s_sps}). 
     423(\autoref{fig:DOM_z_zps_s_sps}) or an envelop bathymetry can be defined (\autoref{fig:DOM_z_zps_s_sps}). 
    424424The namelist parameter \np{rn\_rmax} determines the slope at which 
    425425the terrain-following coordinate intersects the sea bed and becomes a pseudo z-coordinate. 
     
    436436\[ 
    437437  z = s_{min} + C (s) (H - s_{min}) 
    438   % \label{eq:SH94_1} 
     438  % \label{eq:DOMCFG_SH94_1} 
    439439\] 
    440440 
     
    458458         + b       \frac{\tanh \lt[ \theta \lt(s + \frac{1}{2} \rt) \rt] -   \tanh \lt( \frac{\theta}{2} \rt)} 
    459459                        {                                                  2 \tanh \lt( \frac{\theta}{2} \rt)} 
    460  \label{eq:SH94_2} 
     460 \label{eq:DOMCFG_SH94_2} 
    461461\] 
    462462 
     
    466466    \includegraphics[width=\textwidth]{Fig_sco_function} 
    467467    \caption{ 
    468       \protect\label{fig:sco_function} 
     468      \protect\label{fig:DOMCFG_sco_function} 
    469469      Examples of the stretching function applied to a seamount; 
    470470      from left to right: surface, surface and bottom, and bottom intensified resolutions 
     
    478478bottom control parameters such that $0 \leqslant \theta \leqslant 20$, and $0 \leqslant b \leqslant 1$. 
    479479$b$ has been designed to allow surface and/or bottom increase of the vertical resolution 
    480 (\autoref{fig:sco_function}). 
     480(\autoref{fig:DOMCFG_sco_function}). 
    481481 
    482482Another example has been provided at version 3.5 (\np{ln\_s\_SF12}) that allows a fixed surface resolution in 
     
    486486\begin{equation} 
    487487  z = - \gamma h \quad \text{with} \quad 0 \leq \gamma \leq 1 
    488   % \label{eq:z} 
     488  % \label{eq:DOMCFG_z} 
    489489\end{equation} 
    490490 
     
    524524    For clarity every third coordinate surface is shown. 
    525525  } 
    526   \label{fig:fig_compare_coordinates_surface} 
     526  \label{fig:DOMCFG_fig_compare_coordinates_surface} 
    527527\end{figure} 
    528528 % >>>>>>>>>>>>>>>>>>>>>>>>>>>> 
  • NEMO/trunk/doc/latex/NEMO/subfiles/apdx_algos.tex

    r11529 r11543  
    66% ================================================================ 
    77\chapter{Note on some algorithms} 
    8 \label{apdx:E} 
     8\label{apdx:ALGOS} 
    99 
    1010\chaptertoc 
     
    1212\newpage 
    1313 
    14 This appendix some on going consideration on algorithms used or planned to be used in \NEMO.  
     14This appendix some on going consideration on algorithms used or planned to be used in \NEMO. 
    1515 
    1616% ------------------------------------------------------------------------------------------------------------- 
    17 %        UBS scheme   
     17%        UBS scheme 
    1818% ------------------------------------------------------------------------------------------------------------- 
    1919\section{Upstream Biased Scheme (UBS) (\protect\np{ln\_traadv\_ubs}\forcode{ = .true.})} 
     
    4545a constant i-grid spacing ($\Delta i=1$). 
    4646 
    47 Alternative choice: introduce the scale factors:   
     47Alternative choice: introduce the scale factors: 
    4848$\tau "_i =\frac{e_{1T}}{e_{2T}\,e_{3T}}\delta_i \left[ \frac{e_{2u} e_{3u} }{e_{1u} }\delta_{i+1/2}[\tau] \right]$. 
    4949 
     
    5454It is not a \emph{positive} scheme meaning false extrema are permitted but 
    5555the amplitude of such are significantly reduced over the centred second order method. 
    56 Nevertheless it is not recommended to apply it to a passive tracer that requires positivity.  
     56Nevertheless it is not recommended to apply it to a passive tracer that requires positivity. 
    5757 
    5858The intrinsic diffusion of UBS makes its use risky in the vertical direction where 
     
    6161\np{ln\_traadv\_ubs}\forcode{ = .true.}. 
    6262 
    63 For stability reasons, in \autoref{eq:tra_adv_ubs}, the first term which corresponds to 
     63For stability reasons, in \autoref{eq:TRA_adv_ubs}, the first term which corresponds to 
    6464a second order centred scheme is evaluated using the \textit{now} velocity (centred in time) while 
    6565the second term which is the diffusive part of the scheme, is evaluated using the \textit{before} velocity 
     
    6767This is discussed by \citet{webb.de-cuevas.ea_JAOT98} in the context of the Quick advection scheme. 
    6868UBS and QUICK schemes only differ by one coefficient. 
    69 Substituting 1/6 with 1/8 in (\autoref{eq:tra_adv_ubs}) leads to the QUICK advection scheme \citep{webb.de-cuevas.ea_JAOT98}. 
     69Substituting 1/6 with 1/8 in (\autoref{eq:TRA_adv_ubs}) leads to the QUICK advection scheme \citep{webb.de-cuevas.ea_JAOT98}. 
    7070This option is not available through a namelist parameter, since the 1/6 coefficient is hard coded. 
    7171Nevertheless it is quite easy to make the substitution in \mdl{traadv\_ubs} module and obtain a QUICK scheme. 
     
    7575Computer time can be saved by using a time-splitting technique on vertical advection. 
    7676This possibility have been implemented and validated in ORCA05-L301. 
    77 It is not currently offered in the current reference version.  
     77It is not currently offered in the current reference version. 
    7878 
    7979NB 2: In a forthcoming release four options will be proposed for the vertical component used in the UBS scheme. 
     
    8383The $3^{rd}$ case has dispersion properties similar to an eight-order accurate conventional scheme. 
    8484 
    85 NB 3: It is straight forward to rewrite \autoref{eq:tra_adv_ubs} as follows: 
     85NB 3: It is straight forward to rewrite \autoref{eq:TRA_adv_ubs} as follows: 
    8686\begin{equation} 
    8787  \label{eq:tra_adv_ubs2} 
     
    9393  \right. 
    9494\end{equation} 
    95 or equivalently  
     95or equivalently 
    9696\begin{equation} 
    9797  \label{eq:tra_adv_ubs2} 
     
    102102  \end{split} 
    103103\end{equation} 
    104 \autoref{eq:tra_adv_ubs2} has several advantages. 
     104\autoref{eq:TRA_adv_ubs2} has several advantages. 
    105105First it clearly evidences that the UBS scheme is based on the fourth order scheme to which 
    106106is added an upstream biased diffusive term. 
    107107Second, this emphasises that the $4^{th}$ order part have to be evaluated at \emph{now} time step, 
    108 not only the $2^{th}$ order part as stated above using \autoref{eq:tra_adv_ubs}. 
     108not only the $2^{th}$ order part as stated above using \autoref{eq:TRA_adv_ubs}. 
    109109Third, the diffusive term is in fact a biharmonic operator with a eddy coefficient which 
    110110is simply proportional to the velocity. 
     
    134134  \end{split} 
    135135\end{equation} 
    136 with ${A_u^{lT}}^2 = \frac{1}{12} {e_{1u}}^3\ |u|$,  
     136with ${A_u^{lT}}^2 = \frac{1}{12} {e_{1u}}^3\ |u|$, 
    137137\ie\ $A_u^{lT} = \frac{1}{\sqrt{12}} \,e_{1u}\ \sqrt{ e_{1u}\,|u|\,}$ 
    138138it comes: 
     
    189189 
    190190% ------------------------------------------------------------------------------------------------------------- 
    191 %        Leap-Frog energetic   
     191%        Leap-Frog energetic 
    192192% ------------------------------------------------------------------------------------------------------------- 
    193193\section{Leapfrog energetic} 
     
    214214  \equiv \frac{1}{\rdt} \overline{ \delta_{t+\rdt/2}[q]}^{\,t} 
    215215  =         \frac{q^{t+\rdt}-q^{t-\rdt}}{2\rdt} 
    216 \]  
     216\] 
    217217Note that \autoref{chap:LF} shows that the leapfrog time step is $\rdt$, 
    218218not $2\rdt$ as it can be found sometimes in literature. 
     
    226226\] 
    227227is satisfied in discrete form. 
    228 Indeed,  
     228Indeed, 
    229229\[ 
    230230  \begin{split} 
     
    240240\] 
    241241NB here pb of boundary condition when applying the adjoint! 
    242 In space, setting to 0 the quantity in land area is sufficient to get rid of the boundary condition  
     242In space, setting to 0 the quantity in land area is sufficient to get rid of the boundary condition 
    243243(equivalently of the boundary value of the integration by part). 
    244244In time this boundary condition is not physical and \textbf{add something here!!!} 
    245245 
    246246% ================================================================ 
    247 % Iso-neutral diffusion :  
     247% Iso-neutral diffusion : 
    248248% ================================================================ 
    249249 
     
    251251 
    252252% ================================================================ 
    253 % Griffies' iso-neutral diffusion operator :  
     253% Griffies' iso-neutral diffusion operator : 
    254254% ================================================================ 
    255255\subsection{Griffies iso-neutral diffusion operator} 
     
    258258but is formulated within the \NEMO\ framework 
    259259(\ie\ using scale factors rather than grid-size and having a position of $T$-points that 
    260 is not necessary in the middle of vertical velocity points, see \autoref{fig:zgr_e3}). 
    261  
    262 In the formulation \autoref{eq:tra_ldf_iso} introduced in 1995 in OPA, the ancestor of \NEMO, 
     260is not necessary in the middle of vertical velocity points, see \autoref{fig:DOM_zgr_e3}). 
     261 
     262In the formulation \autoref{eq:TRA_ldf_iso} introduced in 1995 in OPA, the ancestor of \NEMO, 
    263263the off-diagonal terms of the small angle diffusion tensor contain several double spatial averages of a gradient, 
    264264for example $\overline{\overline{\delta_k \cdot}}^{\,i,k}$. 
     
    318318%>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> 
    319319 
    320 The four iso-neutral fluxes associated with the triads are defined at $T$-point.  
     320The four iso-neutral fluxes associated with the triads are defined at $T$-point. 
    321321They take the following expression: 
    322322\begin{flalign*} 
     
    332332 
    333333The resulting iso-neutral fluxes at $u$- and $w$-points are then given by 
    334 the sum of the fluxes that cross the $u$- and $w$-face (\autoref{fig:ISO_triad}): 
     334the sum of the fluxes that cross the $u$- and $w$-face (\autoref{fig:TRIADS_ISO_triad}): 
    335335\begin{flalign} 
    336336  \label{eq:iso_flux} 
     
    369369    + \delta_{k} \left[ {_i^{k+1/2-k_p} {\mathbb{F}_w}_{i_p}^{k_p}} \right]   \right\} 
    370370\end{equation} 
    371 where $b_T= e_{1T}\,e_{2T}\,e_{3T}$ is the volume of $T$-cells.  
     371where $b_T= e_{1T}\,e_{2T}\,e_{3T}$ is the volume of $T$-cells. 
    372372 
    373373This expression of the iso-neutral diffusion has been chosen in order to satisfy the following six properties: 
     
    448448 
    449449% ================================================================ 
    450 % Skew flux formulation for Eddy Induced Velocity :  
     450% Skew flux formulation for Eddy Induced Velocity : 
    451451% ================================================================ 
    452452\subsection{Eddy induced velocity and skew flux formulation} 
     
    457457the formulation of which depends on the slopes of iso-neutral surfaces. 
    458458Contrary to the case of iso-neutral mixing, the slopes used here are referenced to the geopotential surfaces, 
    459 \ie\ \autoref{eq:ldfslp_geo} is used in $z$-coordinate, 
    460 and the sum \autoref{eq:ldfslp_geo} + \autoref{eq:ldfslp_iso} in $z^*$ or $s$-coordinates.  
    461  
    462 The eddy induced velocity is given by:  
     459\ie\ \autoref{eq:LDF_slp_geo} is used in $z$-coordinate, 
     460and the sum \autoref{eq:LDF_slp_geo} + \autoref{eq:LDF_slp_iso} in $z^*$ or $s$-coordinates. 
     461 
     462The eddy induced velocity is given by: 
    463463\begin{equation} 
    464464  \label{eq:eiv_v} 
     
    484484(see \autoref{sec:TRA_adv}) and not just a $2^{nd}$ order advection scheme. 
    485485This is particularly useful for passive tracers where 
    486 \emph{positivity} of the advection scheme is of paramount importance.  
    487 % give here the expression using the triads. It is different from the one given in \autoref{eq:ldfeiv} 
     486\emph{positivity} of the advection scheme is of paramount importance. 
     487% give here the expression using the triads. It is different from the one given in \autoref{eq:LDF_eiv} 
    488488% see just below a copy of this equation: 
    489489%\begin{equation} \label{eq:ldfeiv} 
     
    593593  \right) 
    594594\end{equation} 
    595 Note that \autoref{eq:eiv_skew} is valid in $z$-coordinate with or without partial cells.  
     595Note that \autoref{eq:eiv_skew} is valid in $z$-coordinate with or without partial cells. 
    596596In $z^*$ or $s$-coordinate, the slope between the level and the geopotential surfaces must be added to 
    597 $\mathbb{R}$ for the discret form to be exact.  
     597$\mathbb{R}$ for the discret form to be exact. 
    598598 
    599599Such a choice of discretisation is consistent with the iso-neutral operator as 
     
    604604$\ $\newpage      %force an empty line 
    605605% ================================================================ 
    606 % Discrete Invariants of the iso-neutral diffrusion  
     606% Discrete Invariants of the iso-neutral diffrusion 
    607607% ================================================================ 
    608608\subsection{Discrete invariants of the iso-neutral diffrusion} 
    609609\label{subsec:Gf_operator} 
    610610 
    611 Demonstration of the decrease of the tracer variance in the (\textbf{i},\textbf{j}) plane.  
     611Demonstration of the decrease of the tracer variance in the (\textbf{i},\textbf{j}) plane. 
    612612 
    613613This part will be moved in an Appendix. 
     
    617617  \int_D  D_l^T \; T \;dv   \leq 0 
    618618\] 
    619 The discrete form of its left hand side is obtained using \autoref{eq:iso_flux} 
     619The discrete form of its left hand side is obtained using \autoref{eq:TRIADS_iso_flux} 
    620620 
    621621\begin{align*} 
     
    740740        \right\} 
    741741        \quad   \leq 0 
    742 \end{align*}  
     742\end{align*} 
    743743The last inequality is obviously obtained as we succeed in obtaining a negative summation of square quantities. 
    744744 
     
    764764                             % 
    765765                           &\equiv  \sum_{i,k} \left\{ D_l^S \ T \ b_T \right\} 
    766 \end{align*}  
     766\end{align*} 
    767767This means that the iso-neutral operator is self-adjoint. 
    768768There is no need to develop a specific to obtain it. 
     
    776776\label{subsec:eiv_skew} 
    777777 
    778 Demonstration for the conservation of the tracer variance in the (\textbf{i},\textbf{j}) plane.  
     778Demonstration for the conservation of the tracer variance in the (\textbf{i},\textbf{j}) plane. 
    779779 
    780780This have to be moved in an Appendix. 
     
    830830    &{\ \ \;_i^k  \mathbb{R}_{+1/2}^{+1/2}}   &\delta_{i+1/2}[T^{k\ \ \ \:}]  &\delta_{k+1/2}[T_{i}] 
    831831    &\Bigr\}  \\ 
    832   \end{matrix}    
     832  \end{matrix} 
    833833\end{align*} 
    834834The two terms associated with the triad ${_i^k \mathbb{R}_{+1/2}^{+1/2}}$ are the same but of opposite signs, 
    835 they cancel out.  
     835they cancel out. 
    836836Exactly the same thing occurs for the triad ${_i^k \mathbb{R}_{-1/2}^{-1/2}}$. 
    837837The two terms associated with the triad ${_i^k \mathbb{R}_{+1/2}^{-1/2}}$ are the same but both of opposite signs and 
  • NEMO/trunk/doc/latex/NEMO/subfiles/apdx_diff_opers.tex

    r11529 r11543  
    55% Chapter Appendix B : Diffusive Operators 
    66% ================================================================ 
    7 \chapter{Appendix B : Diffusive Operators} 
    8 \label{apdx:B} 
     7\chapter{Diffusive Operators} 
     8\label{apdx:DIFFOPERS} 
    99 
    1010\chaptertoc 
     
    1616% ================================================================ 
    1717\section{Horizontal/Vertical $2^{nd}$ order tracer diffusive operators} 
    18 \label{sec:B_1} 
     18\label{sec:DIFFOPERS_1} 
    1919 
    2020\subsubsection*{In z-coordinates} 
     
    2222In $z$-coordinates, the horizontal/vertical second order tracer diffusion operator is given by: 
    2323\begin{align} 
    24   \label{apdx:B1} 
     24  \label{eq:DIFFOPERS_1} 
    2525  &D^T = \frac{1}{e_1 \, e_2}      \left[ 
    2626    \left. \frac{\partial}{\partial i} \left(   \frac{e_2}{e_1}A^{lT} \;\left. \frac{\partial T}{\partial i} \right|_z   \right)   \right|_z      \right. 
     
    3232\subsubsection*{In generalized vertical coordinates} 
    3333 
    34 In $s$-coordinates, we defined the slopes of $s$-surfaces, $\sigma_1$ and $\sigma_2$ by \autoref{apdx:A_s_slope} and 
     34In $s$-coordinates, we defined the slopes of $s$-surfaces, $\sigma_1$ and $\sigma_2$ by \autoref{eq:SCOORD_s_slope} and 
    3535the vertical/horizontal ratio of diffusion coefficient by $\epsilon = A^{vT} / A^{lT}$. 
    3636The diffusion operator is given by: 
    3737 
    3838\begin{equation} 
    39   \label{apdx:B2} 
     39  \label{eq:DIFFOPERS_2} 
    4040  D^T = \left. \nabla \right|_s \cdot 
    4141  \left[ A^{lT} \;\Re \cdot \left. \nabla \right|_s T  \right] \\ 
     
    5454  \begin{array}{*{20}l} 
    5555    D^T= \frac{1}{e_1\,e_2\,e_3 } & \left\{ \quad \quad \frac{\partial }{\partial i}  \left. \left[  e_2\,e_3 \, A^{lT} 
    56                                \left( \  \frac{1}{e_1}\; \left. \frac{\partial T}{\partial i} \right|_s  
     56                               \left( \  \frac{1}{e_1}\; \left. \frac{\partial T}{\partial i} \right|_s 
    5757                                       -\frac{\sigma_1 }{e_3 } \; \frac{\partial T}{\partial s} \right) \right]  \right|_s  \right. \\ 
    5858        &  \quad \  +   \            \left.   \frac{\partial }{\partial j}  \left. \left[  e_1\,e_3 \, A^{lT} 
    59                                \left( \ \frac{1}{e_2 }\; \left. \frac{\partial T}{\partial j} \right|_s  
     59                               \left( \ \frac{1}{e_2 }\; \left. \frac{\partial T}{\partial j} \right|_s 
    6060                                       -\frac{\sigma_2 }{e_3 } \; \frac{\partial T}{\partial s} \right) \right]  \right|_s  \right. \\ 
    61         &  \quad \  +   \           \left.  e_1\,e_2\, \frac{\partial }{\partial s}  \left[ A^{lT} \; \left(  
    62                      -\frac{\sigma_1 }{e_1 } \; \left. \frac{\partial T}{\partial i} \right|_s  
    63                      -\frac{\sigma_2 }{e_2 } \; \left. \frac{\partial T}{\partial j} \right|_s  
     61        &  \quad \  +   \           \left.  e_1\,e_2\, \frac{\partial }{\partial s}  \left[ A^{lT} \; \left( 
     62                     -\frac{\sigma_1 }{e_1 } \; \left. \frac{\partial T}{\partial i} \right|_s 
     63                     -\frac{\sigma_2 }{e_2 } \; \left. \frac{\partial T}{\partial j} \right|_s 
    6464                          +\left( \varepsilon +\sigma_1^2+\sigma_2 ^2 \right) \; \frac{1}{e_3 } \; \frac{\partial T}{\partial s} \right) \; \right] \;  \right\} . 
    6565  \end{array} 
     
    6767\end{align*} 
    6868 
    69 \autoref{apdx:B2} is obtained from \autoref{apdx:B1} without any additional assumption. 
     69\autoref{eq:DIFFOPERS_2} is obtained from \autoref{eq:DIFFOPERS_1} without any additional assumption. 
    7070Indeed, for the special case $k=z$ and thus $e_3 =1$, 
    71 we introduce an arbitrary vertical coordinate $s = s (i,j,z)$ as in \autoref{apdx:A} and 
    72 use \autoref{apdx:A_s_slope} and \autoref{apdx:A_s_chain_rule}. 
    73 Since no cross horizontal derivative $\partial _i \partial _j $ appears in \autoref{apdx:B1}, 
     71we introduce an arbitrary vertical coordinate $s = s (i,j,z)$ as in \autoref{apdx:SCOORD} and 
     72use \autoref{eq:SCOORD_s_slope} and \autoref{eq:SCOORD_s_chain_rule}. 
     73Since no cross horizontal derivative $\partial _i \partial _j $ appears in \autoref{eq:DIFFOPERS_1}, 
    7474the ($i$,$z$) and ($j$,$z$) planes are independent. 
    7575The derivation can then be demonstrated for the ($i$,$z$)~$\to$~($j$,$s$) transformation without 
     
    160160% ================================================================ 
    161161\section{Iso/Diapycnal $2^{nd}$ order tracer diffusive operators} 
    162 \label{sec:B_2} 
     162\label{sec:DIFFOPERS_2} 
    163163 
    164164\subsubsection*{In z-coordinates} 
     
    170170 
    171171\begin{equation} 
    172   \label{apdx:B3} 
     172  \label{eq:DIFFOPERS_3} 
    173173  \textbf {A}_{\textbf I} = \frac{A^{lT}}{\left( {1+a_1 ^2+a_2 ^2} \right)} 
    174174  \left[ {{ 
     
    193193 
    194194In practice, $\epsilon$ is small and isopycnal slopes are generally less than $10^{-2}$ in the ocean, 
    195 so $\textbf {A}_{\textbf I}$ can be simplified appreciably \citep{cox_OM87}. Keeping leading order terms\footnote{Apart from the (1,0)  
     195so $\textbf {A}_{\textbf I}$ can be simplified appreciably \citep{cox_OM87}. Keeping leading order terms\footnote{Apart from the (1,0) 
    196196and (0,1) elements which are set to zero. See \citet{griffies_bk04}, section 14.1.4.1 for a discussion of this point.}: 
    197197\begin{subequations} 
    198   \label{apdx:B4} 
     198  \label{eq:DIFFOPERS_4} 
    199199  \begin{equation} 
    200     \label{apdx:B4a} 
     200    \label{eq:DIFFOPERS_4a} 
    201201    {\textbf{A}_{\textbf{I}}} \approx A^{lT}\;\Re\;\text{where} \;\Re = 
    202202    \left[ {{ 
     
    210210  and the iso/dianeutral diffusive operator in $z$-coordinates is then 
    211211  \begin{equation} 
    212     \label{apdx:B4b} 
     212    \label{eq:DIFFOPERS_4b} 
    213213    D^T = \left. \nabla \right|_z \cdot 
    214214    \left[ A^{lT} \;\Re \cdot \left. \nabla \right|_z T  \right]. \\ 
     
    216216\end{subequations} 
    217217 
    218 Physically, the full tensor \autoref{apdx:B3} represents strong isoneutral diffusion on a plane parallel to 
     218Physically, the full tensor \autoref{eq:DIFFOPERS_3} represents strong isoneutral diffusion on a plane parallel to 
    219219the isoneutral surface and weak dianeutral diffusion perpendicular to this plane. 
    220220However, 
    221 the approximate `weak-slope' tensor \autoref{apdx:B4a} represents strong diffusion along the isoneutral surface, 
     221the approximate `weak-slope' tensor \autoref{eq:DIFFOPERS_4a} represents strong diffusion along the isoneutral surface, 
    222222with weak \emph{vertical} diffusion -- the principal axes of the tensor are no longer orthogonal. 
    223223This simplification also decouples the ($i$,$z$) and ($j$,$z$) planes of the tensor. 
    224 The weak-slope operator therefore takes the same form, \autoref{apdx:B4}, as \autoref{apdx:B2}, 
     224The weak-slope operator therefore takes the same form, \autoref{eq:DIFFOPERS_4}, as \autoref{eq:DIFFOPERS_2}, 
    225225the diffusion operator for geopotential diffusion written in non-orthogonal $i,j,s$-coordinates. 
    226226Written out explicitly, 
    227227 
    228228\begin{multline} 
    229   \label{apdx:B_ldfiso} 
     229  \label{eq:DIFFOPERS_ldfiso} 
    230230  D^T=\frac{1}{e_1 e_2 }\left\{ 
    231231    {\;\frac{\partial }{\partial i}\left[ {A_h \left( {\frac{e_2}{e_1}\frac{\partial T}{\partial i}-a_1 \frac{e_2}{e_3}\frac{\partial T}{\partial k}} \right)} \right]} 
     
    234234\end{multline} 
    235235 
    236 The isopycnal diffusion operator \autoref{apdx:B4}, 
    237 \autoref{apdx:B_ldfiso} conserves tracer quantity and dissipates its square. 
    238 As \autoref{apdx:B4} is the divergence of a flux, the demonstration of the first property is trivial, providing that the flux normal to the boundary is zero  
     236The isopycnal diffusion operator \autoref{eq:DIFFOPERS_4}, 
     237\autoref{eq:DIFFOPERS_ldfiso} conserves tracer quantity and dissipates its square. 
     238As \autoref{eq:DIFFOPERS_4} is the divergence of a flux, the demonstration of the first property is trivial, providing that the flux normal to the boundary is zero 
    239239(as it is when $A_h$ is zero at the boundary). Let us demonstrate the second one: 
    240240\[ 
     
    256256             j}-a_2 \frac{\partial T}{\partial k}} \right)^2} 
    257257             +\varepsilon \left(\frac{\partial T}{\partial k}\right) ^2\right]      \\ 
    258            & \geq 0 .  
     258           & \geq 0 . 
    259259  \end{array} 
    260260             } 
     
    265265\subsubsection*{In generalized vertical coordinates} 
    266266 
    267 Because the weak-slope operator \autoref{apdx:B4}, 
    268 \autoref{apdx:B_ldfiso} is decoupled in the ($i$,$z$) and ($j$,$z$) planes, 
     267Because the weak-slope operator \autoref{eq:DIFFOPERS_4}, 
     268\autoref{eq:DIFFOPERS_ldfiso} is decoupled in the ($i$,$z$) and ($j$,$z$) planes, 
    269269it may be transformed into generalized $s$-coordinates in the same way as 
    270 \autoref{sec:B_1} was transformed into \autoref{sec:B_2}. 
     270\autoref{sec:DIFFOPERS_1} was transformed into \autoref{sec:DIFFOPERS_2}. 
    271271The resulting operator then takes the simple form 
    272272 
    273273\begin{equation} 
    274   \label{apdx:B_ldfiso_s} 
     274  \label{eq:DIFFOPERS_ldfiso_s} 
    275275  D^T = \left. \nabla \right|_s \cdot 
    276276  \left[ A^{lT} \;\Re \cdot \left. \nabla \right|_s T  \right] \\ 
     
    295295\] 
    296296 
    297 To prove \autoref{apdx:B_ldfiso_s} by direct re-expression of \autoref{apdx:B_ldfiso} is straightforward, but laborious. 
    298 An easier way is first to note (by reversing the derivation of \autoref{sec:B_2} from \autoref{sec:B_1} ) that 
     297To prove \autoref{eq:DIFFOPERS_ldfiso_s} by direct re-expression of \autoref{eq:DIFFOPERS_ldfiso} is straightforward, but laborious. 
     298An easier way is first to note (by reversing the derivation of \autoref{sec:DIFFOPERS_2} from \autoref{sec:DIFFOPERS_1} ) that 
    299299the weak-slope operator may be \emph{exactly} reexpressed in non-orthogonal $i,j,\rho$-coordinates as 
    300300 
    301301\begin{equation} 
    302   \label{apdx:B5} 
     302  \label{eq:DIFFOPERS_5} 
    303303  D^T = \left. \nabla \right|_\rho \cdot 
    304304  \left[ A^{lT} \;\Re \cdot \left. \nabla \right|_\rho T  \right] \\ 
     
    312312\end{equation} 
    313313Then direct transformation from $i,j,\rho$-coordinates to $i,j,s$-coordinates gives 
    314 \autoref{apdx:B_ldfiso_s} immediately. 
     314\autoref{eq:DIFFOPERS_ldfiso_s} immediately. 
    315315 
    316316Note that the weak-slope approximation is only made in transforming from 
     
    318318The further transformation into $i,j,s$-coordinates is exact, whatever the steepness of the $s$-surfaces, 
    319319in the same way as the transformation of horizontal/vertical Laplacian diffusion in $z$-coordinates in 
    320 \autoref{sec:B_1} onto $s$-coordinates is exact, however steep the $s$-surfaces. 
     320\autoref{sec:DIFFOPERS_1} onto $s$-coordinates is exact, however steep the $s$-surfaces. 
    321321 
    322322 
     
    325325% ================================================================ 
    326326\section{Lateral/Vertical momentum diffusive operators} 
    327 \label{sec:B_3} 
     327\label{sec:DIFFOPERS_3} 
    328328 
    329329The second order momentum diffusion operator (Laplacian) in $z$-coordinates is found by 
    330 applying \autoref{eq:PE_lap_vector}, the expression for the Laplacian of a vector, 
     330applying \autoref{eq:MB_lap_vector}, the expression for the Laplacian of a vector, 
    331331to the horizontal velocity vector: 
    332332\begin{align*} 
     
    371371  }} \right) 
    372372\end{align*} 
    373 Using \autoref{eq:PE_div}, the definition of the horizontal divergence, 
     373Using \autoref{eq:MB_div}, the definition of the horizontal divergence, 
    374374the third component of the second vector is obviously zero and thus : 
    375375\[ 
    376   \Delta {\textbf{U}}_h = \nabla _h \left( \chi \right) - \nabla _h \times \left( \zeta \textbf{k} \right) + \frac {1}{e_3 } \frac {\partial }{\partial k} \left( {\frac {1}{e_3 } \frac{\partial {\textbf{ U}}_h }{\partial k}} \right) .  
     376  \Delta {\textbf{U}}_h = \nabla _h \left( \chi \right) - \nabla _h \times \left( \zeta \textbf{k} \right) + \frac {1}{e_3 } \frac {\partial }{\partial k} \left( {\frac {1}{e_3 } \frac{\partial {\textbf{ U}}_h }{\partial k}} \right) . 
    377377\] 
    378378 
    379379Note that this operator ensures a full separation between 
    380 the vorticity and horizontal divergence fields (see \autoref{apdx:C}). 
     380the vorticity and horizontal divergence fields (see \autoref{apdx:INVARIANTS}). 
    381381It is only equal to a Laplacian applied to each component in Cartesian coordinates, not on the sphere. 
    382382 
     
    384384the $z$-coordinate therefore takes the following form: 
    385385\begin{equation} 
    386   \label{apdx:B_Lap_U} 
     386  \label{eq:DIFFOPERS_Lap_U} 
    387387  { 
    388388    \textbf{D}}^{\textbf{U}} = 
     
    404404\end{align*} 
    405405 
    406 Note Bene: introducing a rotation in \autoref{apdx:B_Lap_U} does not lead to 
     406Note Bene: introducing a rotation in \autoref{eq:DIFFOPERS_Lap_U} does not lead to 
    407407a useful expression for the iso/diapycnal Laplacian operator in the $z$-coordinate. 
    408408Similarly, we did not found an expression of practical use for 
    409409the geopotential horizontal/vertical Laplacian operator in the $s$-coordinate. 
    410 Generally, \autoref{apdx:B_Lap_U} is used in both $z$- and $s$-coordinate systems, 
     410Generally, \autoref{eq:DIFFOPERS_Lap_U} is used in both $z$- and $s$-coordinate systems, 
    411411that is a Laplacian diffusion is applied on momentum along the coordinate directions. 
    412412 
  • NEMO/trunk/doc/latex/NEMO/subfiles/apdx_invariants.tex

    r11529 r11543  
    66% ================================================================ 
    77\chapter{Discrete Invariants of the Equations} 
    8 \label{apdx:C} 
     8\label{apdx:INVARIANTS} 
    99 
    1010\chaptertoc 
     
    2121% ================================================================ 
    2222\section{Introduction / Notations} 
    23 \label{sec:C.0} 
     23\label{sec:INVARIANTS_0} 
    2424 
    2525Notation used in this appendix in the demonstations: 
     
    7272that is in a more compact form : 
    7373\begin{flalign} 
    74   \label{eq:Q2_flux} 
     74  \label{eq:INVARIANTS_Q2_flux} 
    7575  \partial_t \left( \int_D {\frac{1}{2} Q^2\;dv} \right) 
    7676  =&                   \int_D { \frac{Q}{e_3}  \partial_t \left( e_3 \, Q \right) dv } 
     
    8787that is in a more compact form: 
    8888\begin{flalign} 
    89   \label{eq:Q2_vect} 
     89  \label{eq:INVARIANTS_Q2_vect} 
    9090  \partial_t \left( \int_D {\frac{1}{2} Q^2\;dv} \right) 
    9191  =& \int_D {         Q   \,\partial_t Q  \;dv } 
     
    9797% ================================================================ 
    9898\section{Continuous conservation} 
    99 \label{sec:C.1} 
     99\label{sec:INVARIANTS_1} 
    100100 
    101101The discretization of pimitive equation in $s$-coordinate (\ie\ time and space varying vertical coordinate) 
     
    105105The total energy (\ie\ kinetic plus potential energies) is conserved: 
    106106\begin{flalign} 
    107   \label{eq:Tot_Energy} 
     107  \label{eq:INVARIANTS_Tot_Energy} 
    108108  \partial_t \left( \int_D \left( \frac{1}{2} {\textbf{U}_h}^2 +  \rho \, g \, z \right) \;dv \right)  = & 0 
    109109\end{flalign} 
     
    114114The transformation for the advection term depends on whether the vector invariant form or 
    115115the flux form is used for the momentum equation. 
    116 Using \autoref{eq:Q2_vect} and introducing \autoref{apdx:A_dyn_vect} in 
    117 \autoref{eq:Tot_Energy} for the former form and 
    118 using \autoref{eq:Q2_flux} and introducing \autoref{apdx:A_dyn_flux} in 
    119 \autoref{eq:Tot_Energy} for the latter form leads to: 
    120  
    121 % \label{eq:E_tot} 
     116Using \autoref{eq:INVARIANTS_Q2_vect} and introducing \autoref{eq:SCOORD_dyn_vect} in 
     117\autoref{eq:INVARIANTS_Tot_Energy} for the former form and 
     118using \autoref{eq:INVARIANTS_Q2_flux} and introducing \autoref{eq:SCOORD_dyn_flux} in 
     119\autoref{eq:INVARIANTS_Tot_Energy} for the latter form leads to: 
     120 
     121% \label{eq:INVARIANTS_E_tot} 
    122122advection term (vector invariant form): 
    123123\[ 
    124   % \label{eq:E_tot_vect_vor_1} 
     124  % \label{eq:INVARIANTS_E_tot_vect_vor_1} 
    125125  \int\limits_D  \zeta \; \left( \textbf{k} \times \textbf{U}_h  \right) \cdot \textbf{U}_h  \;  dv   = 0   \\ 
    126126\] 
    127127% 
    128128\[ 
    129   % \label{eq:E_tot_vect_adv_1} 
     129  % \label{eq:INVARIANTS_E_tot_vect_adv_1} 
    130130  \int\limits_D  \textbf{U}_h \cdot \nabla_h \left( \frac{{\textbf{U}_h}^2}{2} \right)     dv 
    131131  + \int\limits_D  \textbf{U}_h \cdot \nabla_z \textbf{U}_h  \;dv 
     
    134134advection term (flux form): 
    135135\[ 
    136   % \label{eq:E_tot_flux_metric} 
     136  % \label{eq:INVARIANTS_E_tot_flux_metric} 
    137137  \int\limits_D  \frac{1} {e_1 e_2 } \left( v \,\partial_i e_2 - u \,\partial_j e_1  \right)\; 
    138138  \left(  \textbf{k} \times \textbf{U}_h  \right) \cdot \textbf{U}_h  \;  dv   = 0 
    139139\] 
    140140\[ 
    141   % \label{eq:E_tot_flux_adv} 
     141  % \label{eq:INVARIANTS_E_tot_flux_adv} 
    142142  \int\limits_D \textbf{U}_h \cdot     \left(                 {{ 
    143143        \begin{array} {*{20}c} 
     
    150150coriolis term 
    151151\[ 
    152   % \label{eq:E_tot_cor} 
     152  % \label{eq:INVARIANTS_E_tot_cor} 
    153153  \int\limits_D  f   \; \left( \textbf{k} \times \textbf{U}_h  \right) \cdot \textbf{U}_h  \;  dv   = 0 
    154154\] 
    155155pressure gradient: 
    156156\[ 
    157   % \label{eq:E_tot_pg_1} 
     157  % \label{eq:INVARIANTS_E_tot_pg_1} 
    158158  - \int\limits_D  \left. \nabla p \right|_z \cdot \textbf{U}_h \;dv 
    159159  = - \int\limits_D \nabla \cdot \left( \rho \,\textbf {U} \right)\;g\;z\;\;dv 
     
    173173 
    174174Vector invariant form: 
    175 % \label{eq:E_tot_vect} 
    176 \[ 
    177   % \label{eq:E_tot_vect_vor_2} 
     175% \label{eq:INVARIANTS_E_tot_vect} 
     176\[ 
     177  % \label{eq:INVARIANTS_E_tot_vect_vor_2} 
    178178  \int\limits_D   \textbf{U}_h \cdot \text{VOR} \;dv   = 0 
    179179\] 
    180180\[ 
    181   % \label{eq:E_tot_vect_adv_2} 
     181  % \label{eq:INVARIANTS_E_tot_vect_adv_2} 
    182182  \int\limits_D  \textbf{U}_h \cdot \text{KEG}  \;dv 
    183183  + \int\limits_D  \textbf{U}_h \cdot \text{ZAD}  \;dv 
     
    185185\] 
    186186\[ 
    187   % \label{eq:E_tot_pg_2} 
     187  % \label{eq:INVARIANTS_E_tot_pg_2} 
    188188  - \int\limits_D  \textbf{U}_h \cdot (\text{HPG}+ \text{SPG}) \;dv 
    189189  = - \int\limits_D \nabla \cdot \left( \rho \,\textbf {U} \right)\;g\;z\;\;dv 
     
    193193Flux form: 
    194194\begin{subequations} 
    195   \label{eq:E_tot_flux} 
     195  \label{eq:INVARIANTS_E_tot_flux} 
    196196  \[ 
    197     % \label{eq:E_tot_flux_metric_2} 
     197    % \label{eq:INVARIANTS_E_tot_flux_metric_2} 
    198198    \int\limits_D  \textbf{U}_h \cdot \text {COR} \;  dv   = 0 
    199199  \] 
    200200  \[ 
    201     % \label{eq:E_tot_flux_adv_2} 
     201    % \label{eq:INVARIANTS_E_tot_flux_adv_2} 
    202202    \int\limits_D \textbf{U}_h \cdot \text{ADV}   \;dv 
    203203    +   \frac{1}{2} \int\limits_D {  {\textbf{U}_h}^2 \frac{1}{e_3} \partial_t e_3  \;dv } =\;0 
    204204  \] 
    205205  \begin{equation} 
    206     \label{eq:E_tot_pg_3} 
     206    \label{eq:INVARIANTS_E_tot_pg_3} 
    207207    - \int\limits_D  \textbf{U}_h \cdot (\text{HPG}+ \text{SPG}) \;dv 
    208208    = - \int\limits_D \nabla \cdot \left( \rho \,\textbf {U} \right)\;g\;z\;\;dv 
     
    211211\end{subequations} 
    212212 
    213 \autoref{eq:E_tot_pg_3} is the balance between the conversion KE to PE and PE to KE. 
    214 Indeed the left hand side of \autoref{eq:E_tot_pg_3} can be transformed as follows: 
     213\autoref{eq:INVARIANTS_E_tot_pg_3} is the balance between the conversion KE to PE and PE to KE. 
     214Indeed the left hand side of \autoref{eq:INVARIANTS_E_tot_pg_3} can be transformed as follows: 
    215215\begin{flalign*} 
    216216  \partial_t  \left( \int\limits_D { \rho \, g \, z  \;dv} \right) 
     
    225225\end{flalign*} 
    226226where the last equality is obtained by noting that the brackets is exactly the expression of $w$, 
    227 the vertical velocity referenced to the fixe $z$-coordinate system (see \autoref{apdx:A_w_s}). 
    228  
    229 The left hand side of \autoref{eq:E_tot_pg_3} can be transformed as follows: 
     227the vertical velocity referenced to the fixe $z$-coordinate system (see \autoref{eq:SCOORD_w_s}). 
     228 
     229The left hand side of \autoref{eq:INVARIANTS_E_tot_pg_3} can be transformed as follows: 
    230230\begin{flalign*} 
    231231  - \int\limits_D  \left. \nabla p \right|_z & \cdot \textbf{U}_h \;dv 
     
    326326% ================================================================ 
    327327\section{Discrete total energy conservation: vector invariant form} 
    328 \label{sec:C.2} 
     328\label{sec:INVARIANTS_2} 
    329329 
    330330% ------------------------------------------------------------------------------------------------------------- 
     
    332332% ------------------------------------------------------------------------------------------------------------- 
    333333\subsection{Total energy conservation} 
    334 \label{subsec:C_KE+PE_vect} 
    335  
    336 The discrete form of the total energy conservation, \autoref{eq:Tot_Energy}, is given by: 
     334\label{subsec:INVARIANTS_KE+PE_vect} 
     335 
     336The discrete form of the total energy conservation, \autoref{eq:INVARIANTS_Tot_Energy}, is given by: 
    337337\begin{flalign*} 
    338338  \partial_t  \left(  \sum\limits_{i,j,k} \biggl\{ \frac{u^2}{2} \,b_u + \frac{v^2}{2}\, b_v +  \rho \, g \, z_t \,b_t  \biggr\} \right) &=0 
     
    340340which in vector invariant forms, it leads to: 
    341341\begin{equation} 
    342   \label{eq:KE+PE_vect_discrete} 
     342  \label{eq:INVARIANTS_KE+PE_vect_discrete} 
    343343  \begin{split} 
    344344    \sum\limits_{i,j,k} \biggl\{   u\,                        \partial_t u         \;b_u 
     
    352352 
    353353Substituting the discrete expression of the time derivative of the velocity either in vector invariant, 
    354 leads to the discrete equivalent of the four equations \autoref{eq:E_tot_flux}. 
     354leads to the discrete equivalent of the four equations \autoref{eq:INVARIANTS_E_tot_flux}. 
    355355 
    356356% ------------------------------------------------------------------------------------------------------------- 
     
    358358% ------------------------------------------------------------------------------------------------------------- 
    359359\subsection{Vorticity term (coriolis + vorticity part of the advection)} 
    360 \label{subsec:C_vor} 
     360\label{subsec:INVARIANTS_vor} 
    361361 
    362362Let $q$, located at $f$-points, be either the relative ($q=\zeta / e_{3f}$), 
     
    367367% ------------------------------------------------------------------------------------------------------------- 
    368368\subsubsection{Vorticity term with ENE scheme (\protect\np{ln\_dynvor\_ene}\forcode{ = .true.})} 
    369 \label{subsec:C_vorENE} 
     369\label{subsec:INVARIANTS_vorENE} 
    370370 
    371371For the ENE scheme, the two components of the vorticity term are given by: 
     
    407407% ------------------------------------------------------------------------------------------------------------- 
    408408\subsubsection{Vorticity term with EEN scheme (\protect\np{ln\_dynvor\_een}\forcode{ = .true.})} 
    409 \label{subsec:C_vorEEN_vect} 
     409\label{subsec:INVARIANTS_vorEEN_vect} 
    410410 
    411411With the EEN scheme, the vorticity terms are represented as: 
    412412\begin{equation} 
    413   \tag{\ref{eq:dynvor_een}} 
     413  \label{eq:INVARIANTS_dynvor_een} 
    414414  \left\{ { 
    415415      \begin{aligned} 
     
    424424and the vorticity triads, ${^i_j}\mathbb{Q}^{i_p}_{j_p}$, defined at $T$-point, are given by: 
    425425\begin{equation} 
    426   \tag{\ref{eq:Q_triads}} 
     426  \label{eq:INVARIANTS_Q_triads} 
    427427  _i^j \mathbb{Q}^{i_p}_{j_p} 
    428428  = \frac{1}{12} \ \left(   q^{i-i_p}_{j+j_p} + q^{i+j_p}_{j+i_p} + q^{i+i_p}_{j-j_p}  \right) 
     
    479479% ------------------------------------------------------------------------------------------------------------- 
    480480\subsubsection{Gradient of kinetic energy / Vertical advection} 
    481 \label{subsec:C_zad} 
     481\label{subsec:INVARIANTS_zad} 
    482482 
    483483The change of Kinetic Energy (KE) due to the vertical advection is exactly balanced by the change of KE due to the horizontal gradient of KE~: 
     
    542542  % 
    543543  \intertext{The first term provides the discrete expression for the vertical advection of momentum (ZAD), 
    544     while the second term corresponds exactly to \autoref{eq:KE+PE_vect_discrete}, therefore:} 
     544    while the second term corresponds exactly to \autoref{eq:INVARIANTS_KE+PE_vect_discrete}, therefore:} 
    545545  \equiv&                   \int\limits_D \textbf{U}_h \cdot \text{ZAD} \;dv 
    546546  + \frac{1}{2} \int_D { {\textbf{U}_h}^2 \frac{1}{e_3} \partial_t  (e_3)  \;dv }    &&&\\ 
     
    578578which is (over-)satified by defining the vertical scale factor as follows: 
    579579\begin{flalign*} 
    580   % \label{eq:e3u-e3v} 
     580  % \label{eq:INVARIANTS_e3u-e3v} 
    581581  e_{3u} = \frac{1}{e_{1u}\,e_{2u}}\;\overline{ e_{1t}^{ }\,e_{2t}^{ }\,e_{3t}^{ } }^{\,i+1/2}    \\ 
    582582  e_{3v} = \frac{1}{e_{1v}\,e_{2v}}\;\overline{ e_{1t}^{ }\,e_{2t}^{ }\,e_{3t}^{ } }^{\,j+1/2} 
     
    590590% ------------------------------------------------------------------------------------------------------------- 
    591591\subsection{Pressure gradient term} 
    592 \label{subsec:C.2.6} 
     592\label{subsec:INVARIANTS_2.6} 
    593593 
    594594\gmcomment{ 
     
    622622  \allowdisplaybreaks 
    623623  \intertext{Using successively \autoref{eq:DOM_di_adj}, \ie\ the skew symmetry property of 
    624     the $\delta$ operator, \autoref{eq:wzv}, the continuity equation, \autoref{eq:dynhpg_sco}, 
     624    the $\delta$ operator, \autoref{eq:DYN_wzv}, the continuity equation, \autoref{eq:DYN_hpg_sco}, 
    625625    the hydrostatic equation in the $s$-coordinate, and $\delta_{k+1/2} \left[ z_t \right] \equiv e_{3w} $, 
    626626    which comes from the definition of $z_t$, it becomes: } 
     
    667667  % 
    668668\end{flalign*} 
    669 The first term is exactly the first term of the right-hand-side of \autoref{eq:KE+PE_vect_discrete}. 
     669The first term is exactly the first term of the right-hand-side of \autoref{eq:INVARIANTS_KE+PE_vect_discrete}. 
    670670It remains to demonstrate that the last term, 
    671671which is obviously a discrete analogue of $\int_D \frac{p}{e_3} \partial_t (e_3)\;dv$ is equal to 
    672 the last term of \autoref{eq:KE+PE_vect_discrete}. 
     672the last term of \autoref{eq:INVARIANTS_KE+PE_vect_discrete}. 
    673673In other words, the following property must be satisfied: 
    674674\begin{flalign*} 
     
    735735% ================================================================ 
    736736\section{Discrete total energy conservation: flux form} 
    737 \label{sec:C.3} 
     737\label{sec:INVARIANTS_3} 
    738738 
    739739% ------------------------------------------------------------------------------------------------------------- 
     
    741741% ------------------------------------------------------------------------------------------------------------- 
    742742\subsection{Total energy conservation} 
    743 \label{subsec:C_KE+PE_flux} 
    744  
    745 The discrete form of the total energy conservation, \autoref{eq:Tot_Energy}, is given by: 
     743\label{subsec:INVARIANTS_KE+PE_flux} 
     744 
     745The discrete form of the total energy conservation, \autoref{eq:INVARIANTS_Tot_Energy}, is given by: 
    746746\begin{flalign*} 
    747747  \partial_t \left(  \sum\limits_{i,j,k} \biggl\{ \frac{u^2}{2} \,b_u + \frac{v^2}{2}\, b_v +  \rho \, g \, z_t \,b_t  \biggr\} \right) &=0  \\ 
     
    765765% ------------------------------------------------------------------------------------------------------------- 
    766766\subsection{Coriolis and advection terms: flux form} 
    767 \label{subsec:C.3.2} 
     767\label{subsec:INVARIANTS_3.2} 
    768768 
    769769% ------------------------------------------------------------------------------------------------------------- 
     
    771771% ------------------------------------------------------------------------------------------------------------- 
    772772\subsubsection{Coriolis plus ``metric'' term} 
    773 \label{subsec:C.3.3} 
     773\label{subsec:INVARIANTS_3.3} 
    774774 
    775775In flux from the vorticity term reduces to a Coriolis term in which 
     
    786786Either the ENE or EEN scheme is then applied to obtain the vorticity term in flux form. 
    787787It therefore conserves the total KE. 
    788 The derivation is the same as for the vorticity term in the vector invariant form (\autoref{subsec:C_vor}). 
     788The derivation is the same as for the vorticity term in the vector invariant form (\autoref{subsec:INVARIANTS_vor}). 
    789789 
    790790% ------------------------------------------------------------------------------------------------------------- 
     
    792792% ------------------------------------------------------------------------------------------------------------- 
    793793\subsubsection{Flux form advection} 
    794 \label{subsec:C.3.4} 
     794\label{subsec:INVARIANTS_3.4} 
    795795 
    796796The flux form operator of the momentum advection is evaluated using 
     
    800800 
    801801\begin{equation} 
    802   \label{eq:C_ADV_KE_flux} 
     802  \label{eq:INVARIANTS_ADV_KE_flux} 
    803803  -  \int_D \textbf{U}_h \cdot     \left(                 {{ 
    804804        \begin{array} {*{20}c} 
     
    863863\] 
    864864which is the discrete form of $ \frac{1}{2} \int_D u \cdot \nabla \cdot \left(   \textbf{U}\,u   \right) \; dv $. 
    865 \autoref{eq:C_ADV_KE_flux} is thus satisfied. 
     865\autoref{eq:INVARIANTS_ADV_KE_flux} is thus satisfied. 
    866866 
    867867When the UBS scheme is used to evaluate the flux form momentum advection, 
     
    873873% ================================================================ 
    874874\section{Discrete enstrophy conservation} 
    875 \label{sec:C.4} 
     875\label{sec:INVARIANTS_4} 
    876876 
    877877% ------------------------------------------------------------------------------------------------------------- 
     
    879879% ------------------------------------------------------------------------------------------------------------- 
    880880\subsubsection{Vorticity term with ENS scheme  (\protect\np{ln\_dynvor\_ens}\forcode{ = .true.})} 
    881 \label{subsec:C_vorENS} 
     881\label{subsec:INVARIANTS_vorENS} 
    882882 
    883883In the ENS scheme, the vorticity term is descretized as follows: 
    884884\begin{equation} 
    885   \tag{\ref{eq:dynvor_ens}} 
     885  \label{eq:INVARIANTS_dynvor_ens} 
    886886  \left\{ 
    887887    \begin{aligned} 
     
    898898it can be shown that: 
    899899\begin{equation} 
    900   \label{eq:C_1.1} 
     900  \label{eq:INVARIANTS_1.1} 
    901901  \int_D {q\,\;{\textbf{k}}\cdot \frac{1} {e_3} \nabla \times \left( {e_3 \, q \;{\textbf{k}}\times {\textbf{U}}_h } \right)\;dv} \equiv 0 
    902902\end{equation} 
    903903where $dv=e_1\,e_2\,e_3 \; di\,dj\,dk$ is the volume element. 
    904 Indeed, using \autoref{eq:dynvor_ens}, 
    905 the discrete form of the right hand side of \autoref{eq:C_1.1} can be transformed as follow: 
     904Indeed, using \autoref{eq:DYN_vor_ens}, 
     905the discrete form of the right hand side of \autoref{eq:INVARIANTS_1.1} can be transformed as follow: 
    906906\begin{flalign*} 
    907907  &\int_D q \,\; \textbf{k} \cdot \frac{1} {e_3 } \nabla \times 
     
    948948% ------------------------------------------------------------------------------------------------------------- 
    949949\subsubsection{Vorticity Term with EEN scheme (\protect\np{ln\_dynvor\_een}\forcode{ = .true.})} 
    950 \label{subsec:C_vorEEN} 
     950\label{subsec:INVARIANTS_vorEEN} 
    951951 
    952952With the EEN scheme, the vorticity terms are represented as: 
    953953\begin{equation} 
    954   \tag{\ref{eq:dynvor_een}} 
     954  \label{eq:INVARIANTS_dynvor_een} 
    955955  \left\{ { 
    956956      \begin{aligned} 
     
    966966and the vorticity triads, ${^i_j}\mathbb{Q}^{i_p}_{j_p}$, defined at $T$-point, are given by: 
    967967\begin{equation} 
    968   \tag{\ref{eq:Q_triads}} 
     968  \tag{\ref{eq:INVARIANTS_Q_triads}} 
    969969  _i^j \mathbb{Q}^{i_p}_{j_p} 
    970970  = \frac{1}{12} \ \left(   q^{i-i_p}_{j+j_p} + q^{i+j_p}_{j+i_p} + q^{i+i_p}_{j-j_p}  \right) 
     
    975975Let consider one of the vorticity triad, for example ${^{i}_j}\mathbb{Q}^{+1/2}_{+1/2} $, 
    976976similar manipulation can be done for the 3 others. 
    977 The discrete form of the right hand side of \autoref{eq:C_1.1} applied to 
     977The discrete form of the right hand side of \autoref{eq:INVARIANTS_1.1} applied to 
    978978this triad only can be transformed as follow: 
    979979 
     
    10211021% ================================================================ 
    10221022\section{Conservation properties on tracers} 
    1023 \label{sec:C.5} 
     1023\label{sec:INVARIANTS_5} 
    10241024 
    10251025All the numerical schemes used in \NEMO\ are written such that the tracer content is conserved by 
     
    10371037% ------------------------------------------------------------------------------------------------------------- 
    10381038\subsection{Advection term} 
    1039 \label{subsec:C.5.1} 
     1039\label{subsec:INVARIANTS_5.1} 
    10401040 
    10411041conservation of a tracer, $T$: 
     
    11031103% ================================================================ 
    11041104\section{Conservation properties on lateral momentum physics} 
    1105 \label{sec:dynldf_properties} 
     1105\label{sec:INVARIANTS_dynldf_properties} 
    11061106 
    11071107The discrete formulation of the horizontal diffusion of momentum ensures 
     
    11241124% ------------------------------------------------------------------------------------------------------------- 
    11251125\subsection{Conservation of potential vorticity} 
    1126 \label{subsec:C.6.1} 
     1126\label{subsec:INVARIANTS_6.1} 
    11271127 
    11281128The lateral momentum diffusion term conserves the potential vorticity: 
     
    11581158% ------------------------------------------------------------------------------------------------------------- 
    11591159\subsection{Dissipation of horizontal kinetic energy} 
    1160 \label{subsec:C.6.2} 
     1160\label{subsec:INVARIANTS_6.2} 
    11611161 
    11621162The lateral momentum diffusion term dissipates the horizontal kinetic energy: 
     
    12101210% ------------------------------------------------------------------------------------------------------------- 
    12111211\subsection{Dissipation of enstrophy} 
    1212 \label{subsec:C.6.3} 
     1212\label{subsec:INVARIANTS_6.3} 
    12131213 
    12141214The lateral momentum diffusion term dissipates the enstrophy when the eddy coefficients are horizontally uniform: 
     
    12341234% ------------------------------------------------------------------------------------------------------------- 
    12351235\subsection{Conservation of horizontal divergence} 
    1236 \label{subsec:C.6.4} 
     1236\label{subsec:INVARIANTS_6.4} 
    12371237 
    12381238When the horizontal divergence of the horizontal diffusion of momentum (discrete sense) is taken, 
     
    12611261% ------------------------------------------------------------------------------------------------------------- 
    12621262\subsection{Dissipation of horizontal divergence variance} 
    1263 \label{subsec:C.6.5} 
     1263\label{subsec:INVARIANTS_6.5} 
    12641264 
    12651265\begin{flalign*} 
     
    12871287% ================================================================ 
    12881288\section{Conservation properties on vertical momentum physics} 
    1289 \label{sec:C.7} 
     1289\label{sec:INVARIANTS_7} 
    12901290 
    12911291As for the lateral momentum physics, 
     
    14581458% ================================================================ 
    14591459\section{Conservation properties on tracer physics} 
    1460 \label{sec:C.8} 
     1460\label{sec:INVARIANTS_8} 
    14611461 
    14621462The numerical schemes used for tracer subgridscale physics are written such that 
     
    14701470% ------------------------------------------------------------------------------------------------------------- 
    14711471\subsection{Conservation of tracers} 
    1472 \label{subsec:C.8.1} 
     1472\label{subsec:INVARIANTS_8.1} 
    14731473 
    14741474constraint of conservation of tracers: 
     
    15031503% ------------------------------------------------------------------------------------------------------------- 
    15041504\subsection{Dissipation of tracer variance} 
    1505 \label{subsec:C.8.2} 
     1505\label{subsec:INVARIANTS_8.2} 
    15061506 
    15071507constraint on the dissipation of tracer variance: 
  • NEMO/trunk/doc/latex/NEMO/subfiles/apdx_s_coord.tex

    r11529 r11543  
    77% ================================================================ 
    88\chapter{Curvilinear $s-$Coordinate Equations} 
    9 \label{apdx:A} 
     9\label{apdx:SCOORD} 
    1010 
    1111\chaptertoc 
     
    2828% ================================================================ 
    2929\section{Chain rule for $s-$coordinates} 
    30 \label{sec:A_chain} 
     30\label{sec:SCOORD_chain} 
    3131 
    3232In order to establish the set of Primitive Equation in curvilinear $s$-coordinates 
    3333(\ie\ an orthogonal curvilinear coordinate in the horizontal and 
    3434an Arbitrary Lagrangian Eulerian (ALE) coordinate in the vertical), 
    35 we start from the set of equations established in \autoref{subsec:PE_zco_Eq} for 
     35we start from the set of equations established in \autoref{subsec:MB_zco_Eq} for 
    3636the special case $k = z$ and thus $e_3 = 1$, 
    3737and we introduce an arbitrary vertical coordinate $a = a(i,j,z,t)$. 
     
    3939the horizontal slope of $s-$surfaces by: 
    4040\begin{equation} 
    41   \label{apdx:A_s_slope} 
     41  \label{eq:SCOORD_s_slope} 
    4242  \sigma_1 =\frac{1}{e_1 } \; \left. {\frac{\partial z}{\partial i}} \right|_s 
    4343  \quad \text{and} \quad 
     
    4646 
    4747The model fields (e.g. pressure $p$) can be viewed as functions of $(i,j,z,t)$ (e.g. $p(i,j,z,t)$) or as 
    48 functions of $(i,j,s,t)$ (e.g. $p(i,j,s,t)$). The symbol $\bullet$ will be used to represent any one of  
    49 these fields.  Any ``infinitesimal'' change in $\bullet$ can be written in two forms:  
    50 \begin{equation} 
    51   \label{apdx:A_s_infin_changes} 
     48functions of $(i,j,s,t)$ (e.g. $p(i,j,s,t)$). The symbol $\bullet$ will be used to represent any one of 
     49these fields.  Any ``infinitesimal'' change in $\bullet$ can be written in two forms: 
     50\begin{equation} 
     51  \label{eq:SCOORD_s_infin_changes} 
    5252  \begin{aligned} 
    53     & \delta \bullet =  \delta i \left. \frac{ \partial \bullet }{\partial i} \right|_{j,s,t}  
    54                 + \delta j \left. \frac{ \partial \bullet }{\partial i} \right|_{i,s,t}  
    55                 + \delta s \left. \frac{ \partial \bullet }{\partial s} \right|_{i,j,t}  
     53    & \delta \bullet =  \delta i \left. \frac{ \partial \bullet }{\partial i} \right|_{j,s,t} 
     54                + \delta j \left. \frac{ \partial \bullet }{\partial i} \right|_{i,s,t} 
     55                + \delta s \left. \frac{ \partial \bullet }{\partial s} \right|_{i,j,t} 
    5656                + \delta t \left. \frac{ \partial \bullet }{\partial t} \right|_{i,j,s} , \\ 
    57     & \delta \bullet =  \delta i \left. \frac{ \partial \bullet }{\partial i} \right|_{j,z,t}  
    58                 + \delta j \left. \frac{ \partial \bullet }{\partial i} \right|_{i,z,t}  
    59                 + \delta z \left. \frac{ \partial \bullet }{\partial z} \right|_{i,j,t}  
     57    & \delta \bullet =  \delta i \left. \frac{ \partial \bullet }{\partial i} \right|_{j,z,t} 
     58                + \delta j \left. \frac{ \partial \bullet }{\partial i} \right|_{i,z,t} 
     59                + \delta z \left. \frac{ \partial \bullet }{\partial z} \right|_{i,j,t} 
    6060                + \delta t \left. \frac{ \partial \bullet }{\partial t} \right|_{i,j,z} . 
    6161  \end{aligned} 
     
    6363Using the first form and considering a change $\delta i$ with $j, z$ and $t$ held constant, shows that 
    6464\begin{equation} 
    65   \label{apdx:A_s_chain_rule} 
     65  \label{eq:SCOORD_s_chain_rule} 
    6666      \left. {\frac{\partial \bullet }{\partial i}} \right|_{j,z,t}  = 
    6767      \left. {\frac{\partial \bullet }{\partial i}} \right|_{j,s,t} 
    68     + \left. {\frac{\partial s       }{\partial i}} \right|_{j,z,t} \;  
    69       \left. {\frac{\partial \bullet }{\partial s}} \right|_{i,j,t} .     
    70 \end{equation} 
    71 The term $\left. \partial s / \partial i \right|_{j,z,t}$ can be related to the slope of constant $s$ surfaces,  
    72 (\autoref{apdx:A_s_slope}), by applying the second of (\autoref{apdx:A_s_infin_changes}) with $\bullet$ set to  
     68    + \left. {\frac{\partial s       }{\partial i}} \right|_{j,z,t} \; 
     69      \left. {\frac{\partial \bullet }{\partial s}} \right|_{i,j,t} . 
     70\end{equation} 
     71The term $\left. \partial s / \partial i \right|_{j,z,t}$ can be related to the slope of constant $s$ surfaces, 
     72(\autoref{eq:SCOORD_s_slope}), by applying the second of (\autoref{eq:SCOORD_s_infin_changes}) with $\bullet$ set to 
    7373$s$ and $j, t$ held constant 
    7474\begin{equation} 
    75 \label{apdx:a_delta_s} 
    76 \delta s|_{j,t} =  
    77          \delta i \left. \frac{ \partial s }{\partial i} \right|_{j,z,t}  
     75\label{eq:SCOORD_delta_s} 
     76\delta s|_{j,t} = 
     77         \delta i \left. \frac{ \partial s }{\partial i} \right|_{j,z,t} 
    7878       + \delta z \left. \frac{ \partial s }{\partial z} \right|_{i,j,t} . 
    7979\end{equation} 
    8080Choosing to look at a direction in the $(i,z)$ plane in which $\delta s = 0$ and using 
    81 (\autoref{apdx:A_s_slope}) we obtain  
    82 \begin{equation} 
    83 \left. \frac{ \partial s }{\partial i} \right|_{j,z,t} =   
     81(\autoref{eq:SCOORD_s_slope}) we obtain 
     82\begin{equation} 
     83\left. \frac{ \partial s }{\partial i} \right|_{j,z,t} = 
    8484         -  \left. \frac{ \partial z }{\partial i} \right|_{j,s,t} \; 
    8585            \left. \frac{ \partial s }{\partial z} \right|_{i,j,t} 
    8686    = - \frac{e_1 }{e_3 }\sigma_1  . 
    87 \label{apdx:a_ds_di_z} 
    88 \end{equation} 
    89 Another identity, similar in form to (\autoref{apdx:a_ds_di_z}), can be derived 
    90 by choosing $\bullet$ to be $s$ and using the second form of (\autoref{apdx:A_s_infin_changes}) to consider 
     87\label{eq:SCOORD_ds_di_z} 
     88\end{equation} 
     89Another identity, similar in form to (\autoref{eq:SCOORD_ds_di_z}), can be derived 
     90by choosing $\bullet$ to be $s$ and using the second form of (\autoref{eq:SCOORD_s_infin_changes}) to consider 
    9191changes in which $i , j$ and $s$ are constant. This shows that 
    9292\begin{equation} 
    93 \label{apdx:A_w_in_s} 
    94 w_s = \left. \frac{ \partial z }{\partial t} \right|_{i,j,s} =   
     93\label{eq:SCOORD_w_in_s} 
     94w_s = \left. \frac{ \partial z }{\partial t} \right|_{i,j,s} = 
    9595- \left. \frac{ \partial z }{\partial s} \right|_{i,j,t} 
    96   \left. \frac{ \partial s }{\partial t} \right|_{i,j,z}  
    97   = - e_3 \left. \frac{ \partial s }{\partial t} \right|_{i,j,z} .  
    98 \end{equation} 
    99  
    100 In what follows, for brevity, indication of the constancy of the $i, j$ and $t$ indices is  
    101 usually omitted. Using the arguments outlined above one can show that the chain rules needed to establish  
     96  \left. \frac{ \partial s }{\partial t} \right|_{i,j,z} 
     97  = - e_3 \left. \frac{ \partial s }{\partial t} \right|_{i,j,z} . 
     98\end{equation} 
     99 
     100In what follows, for brevity, indication of the constancy of the $i, j$ and $t$ indices is 
     101usually omitted. Using the arguments outlined above one can show that the chain rules needed to establish 
    102102the model equations in the curvilinear $s-$coordinate system are: 
    103103\begin{equation} 
    104   \label{apdx:A_s_chain_rule} 
     104  \label{eq:SCOORD_s_chain_rule} 
    105105  \begin{aligned} 
    106106    &\left. {\frac{\partial \bullet }{\partial t}} \right|_z  = 
    107     \left. {\frac{\partial \bullet }{\partial t}} \right|_s  
     107    \left. {\frac{\partial \bullet }{\partial t}} \right|_s 
    108108    + \frac{\partial \bullet }{\partial s}\; \frac{\partial s}{\partial t} , \\ 
    109109    &\left. {\frac{\partial \bullet }{\partial i}} \right|_z  = 
    110110    \left. {\frac{\partial \bullet }{\partial i}} \right|_s 
    111111    +\frac{\partial \bullet }{\partial s}\; \frac{\partial s}{\partial i}= 
    112     \left. {\frac{\partial \bullet }{\partial i}} \right|_s  
     112    \left. {\frac{\partial \bullet }{\partial i}} \right|_s 
    113113    -\frac{e_1 }{e_3 }\sigma_1 \frac{\partial \bullet }{\partial s} , \\ 
    114114    &\left. {\frac{\partial \bullet }{\partial j}} \right|_z  = 
    115     \left. {\frac{\partial \bullet }{\partial j}} \right|_s  
     115    \left. {\frac{\partial \bullet }{\partial j}} \right|_s 
    116116    + \frac{\partial \bullet }{\partial s}\;\frac{\partial s}{\partial j}= 
    117     \left. {\frac{\partial \bullet }{\partial j}} \right|_s  
     117    \left. {\frac{\partial \bullet }{\partial j}} \right|_s 
    118118    - \frac{e_2 }{e_3 }\sigma_2 \frac{\partial \bullet }{\partial s} , \\ 
    119119    &\;\frac{\partial \bullet }{\partial z}  \;\; = \frac{1}{e_3 }\frac{\partial \bullet }{\partial s} . 
     
    126126% ================================================================ 
    127127\section{Continuity equation in $s-$coordinates} 
    128 \label{sec:A_continuity} 
    129  
    130 Using (\autoref{apdx:A_s_chain_rule}) and 
     128\label{sec:SCOORD_continuity} 
     129 
     130Using (\autoref{eq:SCOORD_s_chain_rule}) and 
    131131the fact that the horizontal scale factors $e_1$ and $e_2$ do not depend on the vertical coordinate, 
    132132the divergence of the velocity relative to the ($i$,$j$,$z$) coordinate system is transformed as follows in order to 
     
    189189\end{subequations} 
    190190 
    191 Here, $w$ is the vertical velocity relative to the $z-$coordinate system.  
    192 Using the first form of (\autoref{apdx:A_s_infin_changes})  
    193 and the definitions (\autoref{apdx:A_s_slope}) and (\autoref{apdx:A_w_in_s}) for $\sigma_1$, $\sigma_2$ and  $w_s$, 
     191Here, $w$ is the vertical velocity relative to the $z-$coordinate system. 
     192Using the first form of (\autoref{eq:SCOORD_s_infin_changes}) 
     193and the definitions (\autoref{eq:SCOORD_s_slope}) and (\autoref{eq:SCOORD_w_in_s}) for $\sigma_1$, $\sigma_2$ and  $w_s$, 
    194194one can show that the vertical velocity, $w_p$ of a point 
    195 moving with the horizontal velocity of the fluid along an $s$ surface is given by  
    196 \begin{equation} 
    197 \label{apdx:A_w_p} 
     195moving with the horizontal velocity of the fluid along an $s$ surface is given by 
     196\begin{equation} 
     197\label{eq:SCOORD_w_p} 
    198198\begin{split} 
    199199w_p  = & \left. \frac{ \partial z }{\partial t} \right|_s 
    200      + \frac{u}{e_1} \left. \frac{ \partial z }{\partial i} \right|_s  
     200     + \frac{u}{e_1} \left. \frac{ \partial z }{\partial i} \right|_s 
    201201     + \frac{v}{e_2} \left. \frac{ \partial z }{\partial j} \right|_s \\ 
    202202     = & w_s + u \sigma_1 + v \sigma_2 . 
    203 \end{split}      
     203\end{split} 
    204204\end{equation} 
    205205 The vertical velocity across this surface is denoted by 
    206206\begin{equation} 
    207   \label{apdx:A_w_s} 
    208   \omega  = w - w_p = w - ( w_s + \sigma_1 \,u + \sigma_2 \,v )  .  
    209 \end{equation} 
    210 Hence  
    211 \begin{equation} 
    212 \frac{1}{e_3 } \frac{\partial}{\partial s}   \left[  w -  u\;\sigma_1  - v\;\sigma_2  \right] =  
    213 \frac{1}{e_3 } \frac{\partial}{\partial s} \left[  \omega + w_s \right] =  
    214    \frac{1}{e_3 } \left[ \frac{\partial \omega}{\partial s}  
    215  + \left. \frac{ \partial }{\partial t} \right|_s \frac{\partial z}{\partial s} \right] =  
    216    \frac{1}{e_3 } \frac{\partial \omega}{\partial s} + \frac{1}{e_3 } \left. \frac{ \partial e_3}{\partial t} . \right|_s  
    217 \end{equation} 
    218  
    219 Using (\autoref{apdx:A_w_s}) in our expression for $\nabla \cdot {\mathrm {\mathbf U}}$ we obtain  
     207  \label{eq:SCOORD_w_s} 
     208  \omega  = w - w_p = w - ( w_s + \sigma_1 \,u + \sigma_2 \,v )  . 
     209\end{equation} 
     210Hence 
     211\begin{equation} 
     212\frac{1}{e_3 } \frac{\partial}{\partial s}   \left[  w -  u\;\sigma_1  - v\;\sigma_2  \right] = 
     213\frac{1}{e_3 } \frac{\partial}{\partial s} \left[  \omega + w_s \right] = 
     214   \frac{1}{e_3 } \left[ \frac{\partial \omega}{\partial s} 
     215 + \left. \frac{ \partial }{\partial t} \right|_s \frac{\partial z}{\partial s} \right] = 
     216   \frac{1}{e_3 } \frac{\partial \omega}{\partial s} + \frac{1}{e_3 } \left. \frac{ \partial e_3}{\partial t} . \right|_s 
     217\end{equation} 
     218 
     219Using (\autoref{eq:SCOORD_w_s}) in our expression for $\nabla \cdot {\mathrm {\mathbf U}}$ we obtain 
    220220our final expression for the divergence of the velocity in the curvilinear $s-$coordinate system: 
    221221\begin{equation} 
     
    228228\end{equation} 
    229229 
    230 As a result, the continuity equation \autoref{eq:PE_continuity} in the $s-$coordinates is: 
    231 \begin{equation} 
    232   \label{apdx:A_sco_Continuity} 
     230As a result, the continuity equation \autoref{eq:MB_PE_continuity} in the $s-$coordinates is: 
     231\begin{equation} 
     232  \label{eq:SCOORD_sco_Continuity} 
    233233  \frac{1}{e_3 } \frac{\partial e_3}{\partial t} 
    234234  + \frac{1}{e_1 \,e_2 \,e_3 }\left[ 
     
    245245% ================================================================ 
    246246\section{Momentum equation in $s-$coordinate} 
    247 \label{sec:A_momentum} 
     247\label{sec:SCOORD_momentum} 
    248248 
    249249Here we only consider the first component of the momentum equation, 
     
    252252$\bullet$ \textbf{Total derivative in vector invariant form} 
    253253 
    254 Let us consider \autoref{eq:PE_dyn_vect}, the first component of the momentum equation in the vector invariant form. 
     254Let us consider \autoref{eq:MB_dyn_vect}, the first component of the momentum equation in the vector invariant form. 
    255255Its total $z-$coordinate time derivative, 
    256256$\left. \frac{D u}{D t} \right|_z$ can be transformed as follows in order to obtain 
     
    272272        +  w \;\frac{\partial u}{\partial z}      \\ 
    273273        % 
    274       \intertext{introducing the chain rule (\autoref{apdx:A_s_chain_rule}) } 
     274      \intertext{introducing the chain rule (\autoref{eq:SCOORD_s_chain_rule}) } 
    275275      % 
    276276      &= \left. {\frac{\partial u }{\partial t}} \right|_z 
     
    306306        \; \frac{\partial u}{\partial s} .  \\ 
    307307        % 
    308       \intertext{Introducing $\omega$, the dia-s-surface velocity given by (\autoref{apdx:A_w_s}) } 
     308      \intertext{Introducing $\omega$, the dia-s-surface velocity given by (\autoref{eq:SCOORD_w_s}) } 
    309309      % 
    310310      &= \left. {\frac{\partial u }{\partial t}} \right|_z 
     
    317317\end{subequations} 
    318318% 
    319 Applying the time derivative chain rule (first equation of (\autoref{apdx:A_s_chain_rule})) to $u$ and 
    320 using (\autoref{apdx:A_w_in_s}) provides the expression of the last term of the right hand side, 
     319Applying the time derivative chain rule (first equation of (\autoref{eq:SCOORD_s_chain_rule})) to $u$ and 
     320using (\autoref{eq:SCOORD_w_in_s}) provides the expression of the last term of the right hand side, 
    321321\[ 
    322322  { 
     
    331331\ie\ the total $s-$coordinate time derivative : 
    332332\begin{align} 
    333   \label{apdx:A_sco_Dt_vect} 
     333  \label{eq:SCOORD_sco_Dt_vect} 
    334334  \left. \frac{D u}{D t} \right|_s 
    335335  = \left. {\frac{\partial u }{\partial t}} \right|_s 
    336336  - \left. \zeta \right|_s \;v 
    337337  + \frac{1}{2\,e_1}\left. {\frac{\partial (u^2+v^2)}{\partial i}} \right|_s 
    338   + \frac{1}{e_3 } \omega \;\frac{\partial u}{\partial s} .  
     338  + \frac{1}{e_3 } \omega \;\frac{\partial u}{\partial s} . 
    339339\end{align} 
    340340Therefore, the vector invariant form of the total time derivative has exactly the same mathematical form in 
     
    345345 
    346346Let us start from the total time derivative in the curvilinear $s-$coordinate system we have just establish. 
    347 Following the procedure used to establish (\autoref{eq:PE_flux_form}), it can be transformed into : 
     347Following the procedure used to establish (\autoref{eq:MB_flux_form}), it can be transformed into : 
    348348% \begin{subequations} 
    349349\begin{align*} 
     
    367367\end{align*} 
    368368% 
    369 Introducing the vertical scale factor inside the horizontal derivative of the first two terms  
     369Introducing the vertical scale factor inside the horizontal derivative of the first two terms 
    370370(\ie\ the horizontal divergence), it becomes : 
    371371\begin{align*} 
     
    373373  \begin{array}{*{20}l} 
    374374    % \begin{align*} {\begin{array}{*{20}l} 
    375     %     {\begin{array}{*{20}l} \left. \frac{D u}{D t} \right|_s   
     375    %     {\begin{array}{*{20}l} \left. \frac{D u}{D t} \right|_s 
    376376    &= \left. {\frac{\partial u }{\partial t}} \right|_s 
    377377    &+ \frac{1}{e_1\,e_2\,e_3}  \left(  \frac{\partial( e_2 e_3 \,u^2 )}{\partial i} 
     
    398398     % 
    399399    \intertext {Introducing a more compact form for the divergence of the momentum fluxes, 
    400     and using (\autoref{apdx:A_sco_Continuity}), the $s-$coordinate continuity equation, 
     400    and using (\autoref{eq:SCOORD_sco_Continuity}), the $s-$coordinate continuity equation, 
    401401    it becomes : } 
    402402  % 
     
    410410  } 
    411411\end{align*} 
    412 which leads to the $s-$coordinate flux formulation of the total $s-$coordinate time derivative,  
     412which leads to the $s-$coordinate flux formulation of the total $s-$coordinate time derivative, 
    413413\ie\ the total $s-$coordinate time derivative in flux form: 
    414414\begin{flalign} 
    415   \label{apdx:A_sco_Dt_flux} 
     415  \label{eq:SCOORD_sco_Dt_flux} 
    416416  \left. \frac{D u}{D t} \right|_s   = \frac{1}{e_3}  \left. \frac{\partial ( e_3\,u)}{\partial t} \right|_s 
    417417  + \left.  \nabla \cdot \left(   {{\mathrm {\mathbf U}}\,u}   \right)    \right|_s 
     
    422422It has the same form as in the $z-$coordinate but for 
    423423the vertical scale factor that has appeared inside the time derivative which 
    424 comes from the modification of (\autoref{apdx:A_sco_Continuity}), 
     424comes from the modification of (\autoref{eq:SCOORD_sco_Continuity}), 
    425425the continuity equation. 
    426426 
     
    437437\] 
    438438Applying similar manipulation to the second component and 
    439 replacing $\sigma_1$ and $\sigma_2$ by their expression \autoref{apdx:A_s_slope}, it becomes: 
    440 \begin{equation} 
    441   \label{apdx:A_grad_p_1} 
     439replacing $\sigma_1$ and $\sigma_2$ by their expression \autoref{eq:SCOORD_s_slope}, it becomes: 
     440\begin{equation} 
     441  \label{eq:SCOORD_grad_p_1} 
    442442  \begin{split} 
    443443    -\frac{1}{\rho_o \, e_1 } \left. {\frac{\partial p}{\partial i}} \right|_z 
     
    451451\end{equation} 
    452452 
    453 An additional term appears in (\autoref{apdx:A_grad_p_1}) which accounts for 
     453An additional term appears in (\autoref{eq:SCOORD_grad_p_1}) which accounts for 
    454454the tilt of $s-$surfaces with respect to geopotential $z-$surfaces. 
    455455 
     
    467467Therefore, $p$ and $p_h'$ are linked through: 
    468468\begin{equation} 
    469   \label{apdx:A_pressure} 
     469  \label{eq:SCOORD_pressure} 
    470470  p = \rho_o \; p_h' + \rho_o \, g \, ( \eta - z ) 
    471471\end{equation} 
     
    475475\] 
    476476 
    477 Substituing \autoref{apdx:A_pressure} in \autoref{apdx:A_grad_p_1} and 
     477Substituing \autoref{eq:SCOORD_pressure} in \autoref{eq:SCOORD_grad_p_1} and 
    478478using the definition of the density anomaly it becomes an expression in two parts: 
    479479\begin{equation} 
    480   \label{apdx:A_grad_p_2} 
     480  \label{eq:SCOORD_grad_p_2} 
    481481  \begin{split} 
    482482    -\frac{1}{\rho_o \, e_1 } \left. {\frac{\partial p}{\partial i}} \right|_z 
     
    491491This formulation of the pressure gradient is characterised by the appearance of 
    492492a term depending on the sea surface height only 
    493 (last term on the right hand side of expression \autoref{apdx:A_grad_p_2}). 
     493(last term on the right hand side of expression \autoref{eq:SCOORD_grad_p_2}). 
    494494This term will be loosely termed \textit{surface pressure gradient} whereas 
    495495the first term will be termed the \textit{hydrostatic pressure gradient} by analogy to 
     
    502502The coriolis and forcing terms as well as the the vertical physics remain unchanged as 
    503503they involve neither time nor space derivatives. 
    504 The form of the lateral physics is discussed in \autoref{apdx:B}. 
     504The form of the lateral physics is discussed in \autoref{apdx:DIFFOPERS}. 
    505505 
    506506$\bullet$ \textbf{Full momentum equation} 
     
    510510the one in a curvilinear $z-$coordinate, except for the pressure gradient term: 
    511511\begin{subequations} 
    512   \label{apdx:A_dyn_vect} 
     512  \label{eq:SCOORD_dyn_vect} 
    513513  \begin{multline} 
    514     \label{apdx:A_PE_dyn_vect_u} 
     514    \label{eq:SCOORD_PE_dyn_vect_u} 
    515515    \frac{\partial u}{\partial t}= 
    516516    +   \left( {\zeta +f} \right)\,v 
     
    522522  \end{multline} 
    523523  \begin{multline} 
    524     \label{apdx:A_dyn_vect_v} 
     524    \label{eq:SCOORD_dyn_vect_v} 
    525525    \frac{\partial v}{\partial t}= 
    526526    -   \left( {\zeta +f} \right)\,u 
     
    535535the formulation of both the time derivative and the pressure gradient term: 
    536536\begin{subequations} 
    537   \label{apdx:A_dyn_flux} 
     537  \label{eq:SCOORD_dyn_flux} 
    538538  \begin{multline} 
    539     \label{apdx:A_PE_dyn_flux_u} 
     539    \label{eq:SCOORD_PE_dyn_flux_u} 
    540540    \frac{1}{e_3} \frac{\partial \left(  e_3\,u  \right) }{\partial t} = 
    541541    - \nabla \cdot \left(   {{\mathrm {\mathbf U}}\,u}   \right) 
     
    547547  \end{multline} 
    548548  \begin{multline} 
    549     \label{apdx:A_dyn_flux_v} 
     549    \label{eq:SCOORD_dyn_flux_v} 
    550550    \frac{1}{e_3}\frac{\partial \left(  e_3\,v  \right) }{\partial t}= 
    551551    -  \nabla \cdot \left(   {{\mathrm {\mathbf U}}\,v}   \right) 
     
    554554    -   \frac{1}{e_2 } \left(    \frac{\partial p_h'}{\partial j} + g\; d  \; \frac{\partial z}{\partial j}    \right) 
    555555    -   \frac{g}{e_2 } \frac{\partial \eta}{\partial j} 
    556     +  D_v^{\vect{U}}  +   F_v^{\vect{U}} .  
     556    +  D_v^{\vect{U}}  +   F_v^{\vect{U}} . 
    557557  \end{multline} 
    558558\end{subequations} 
     
    560560hydrostatic pressure and density anomalies, $p_h'$ and $d=( \frac{\rho}{\rho_o}-1 )$: 
    561561\begin{equation} 
    562   \label{apdx:A_dyn_zph} 
     562  \label{eq:SCOORD_dyn_zph} 
    563563  \frac{\partial p_h'}{\partial k} = - d \, g \, e_3 . 
    564564\end{equation} 
     
    569569in particular the pressure gradient. 
    570570By contrast, $\omega$ is not $w$, the third component of the velocity, but the dia-surface velocity component, 
    571 \ie\ the volume flux across the moving $s$-surfaces per unit horizontal area.  
     571\ie\ the volume flux across the moving $s$-surfaces per unit horizontal area. 
    572572 
    573573 
     
    576576% ================================================================ 
    577577\section{Tracer equation} 
    578 \label{sec:A_tracer} 
     578\label{sec:SCOORD_tracer} 
    579579 
    580580The tracer equation is obtained using the same calculation as for the continuity equation and then 
     
    582582 
    583583\begin{multline} 
    584   \label{apdx:A_tracer} 
     584  \label{eq:SCOORD_tracer} 
    585585  \frac{1}{e_3} \frac{\partial \left(  e_3 T  \right)}{\partial t} 
    586586  = -\frac{1}{e_1 \,e_2 \,e_3} 
     
    591591\end{multline} 
    592592 
    593 The expression for the advection term is a straight consequence of (\autoref{apdx:A_sco_Continuity}), 
    594 the expression of the 3D divergence in the $s-$coordinates established above.  
     593The expression for the advection term is a straight consequence of (\autoref{eq:SCOORD_sco_Continuity}), 
     594the expression of the 3D divergence in the $s-$coordinates established above. 
    595595 
    596596\biblio 
  • NEMO/trunk/doc/latex/NEMO/subfiles/apdx_triads.tex

    r11529 r11543  
    1616% ================================================================ 
    1717\chapter{Iso-Neutral Diffusion and Eddy Advection using Triads} 
    18 \label{apdx:triad} 
     18\label{apdx:TRIADS} 
    1919 
    2020\chaptertoc 
     
    4545\begin{description} 
    4646\item[\np{ln\_triad\_iso}] 
    47   See \autoref{sec:taper}. 
     47  See \autoref{sec:TRIADS_taper}. 
    4848  If this is set false (the default), 
    4949  then `iso-neutral' mixing is accomplished within the surface mixed-layer along slopes linearly decreasing with 
    50   depth from the value immediately below the mixed-layer to zero (flat) at the surface (\autoref{sec:lintaper}). 
     50  depth from the value immediately below the mixed-layer to zero (flat) at the surface (\autoref{sec:TRIADS_lintaper}). 
    5151  This is the same treatment as used in the default implementation 
    52   \autoref{subsec:LDF_slp_iso}; \autoref{fig:eiv_slp}. 
     52  \autoref{subsec:LDF_slp_iso}; \autoref{fig:LDF_eiv_slp}. 
    5353  Where \np{ln\_triad\_iso} is set true, 
    5454  the vertical skew flux is further reduced to ensure no vertical buoyancy flux, 
    5555  giving an almost pure horizontal diffusive tracer flux within the mixed layer. 
    56   This is similar to the tapering suggested by \citet{gerdes.koberle.ea_CD91}. See \autoref{subsec:Gerdes-taper} 
     56  This is similar to the tapering suggested by \citet{gerdes.koberle.ea_CD91}. See \autoref{subsec:TRIADS_Gerdes-taper} 
    5757\item[\np{ln\_botmix\_triad}] 
    58   See \autoref{sec:iso_bdry}.  
     58  See \autoref{sec:TRIADS_iso_bdry}. 
    5959  If this is set false (the default) then the lateral diffusive fluxes 
    60   associated with triads partly masked by topography are neglected.  
    61   If it is set true, however, then these lateral diffusive fluxes are applied,  
     60  associated with triads partly masked by topography are neglected. 
     61  If it is set true, however, then these lateral diffusive fluxes are applied, 
    6262  giving smoother bottom tracer fields at the cost of introducing diapycnal mixing. 
    6363\item[\np{rn\_sw\_triad}] 
     
    7171 
    7272\section{Triad formulation of iso-neutral diffusion} 
    73 \label{sec:iso} 
     73\label{sec:TRIADS_iso} 
    7474 
    7575We have implemented into \NEMO\ a scheme inspired by \citet{griffies.gnanadesikan.ea_JPO98}, 
     
    7979 
    8080The iso-neutral second order tracer diffusive operator for small angles between 
    81 iso-neutral surfaces and geopotentials is given by \autoref{eq:iso_tensor_1}: 
     81iso-neutral surfaces and geopotentials is given by \autoref{eq:TRIADS_iso_tensor_1}: 
    8282\begin{subequations} 
    83   \label{eq:iso_tensor_1} 
     83  \label{eq:TRIADS_iso_tensor_1} 
    8484  \begin{equation} 
    8585    D^{lT}=-\nabla \cdot\vect{f}^{lT}\equiv 
     
    9292  \end{equation} 
    9393  \begin{equation} 
    94     \label{eq:iso_tensor_2} 
     94    \label{eq:TRIADS_iso_tensor_2} 
    9595    \mbox{with}\quad \;\;\Re = 
    9696    \begin{pmatrix} 
     
    113113%  {-r_1 } \hfill & {-r_2 } \hfill & {r_1 ^2+r_2 ^2} \hfill \\ 
    114114% \end{array} }} \right) 
    115 Here \autoref{eq:PE_iso_slopes}  
     115Here \autoref{eq:MB_iso_slopes} 
    116116\begin{align*} 
    117117  r_1 &=-\frac{e_3 }{e_1 } \left( \frac{\partial \rho }{\partial i} 
     
    128128We will find it useful to consider the fluxes per unit area in $i,j,k$ space; we write 
    129129\[ 
    130   % \label{eq:Fijk} 
     130  % \label{eq:TRIADS_Fijk} 
    131131  \vect{F}_{\mathrm{iso}}=\left(f_1^{lT}e_2e_3, f_2^{lT}e_1e_3, f_3^{lT}e_1e_2\right). 
    132132\] 
     
    136136 
    137137The off-diagonal terms of the small angle diffusion tensor 
    138 \autoref{eq:iso_tensor_1}, \autoref{eq:iso_tensor_2} produce skew-fluxes along 
     138\autoref{eq:TRIADS_iso_tensor_1}, \autoref{eq:TRIADS_iso_tensor_2} produce skew-fluxes along 
    139139the $i$- and $j$-directions resulting from the vertical tracer gradient: 
    140140\begin{align} 
    141   \label{eq:i13c} 
     141  \label{eq:TRIADS_i13c} 
    142142  f_{13}=&+{A^{lT}} r_1\frac{1}{e_3}\frac{\partial T}{\partial k},\qquad f_{23}=+{A^{lT}} r_2\frac{1}{e_3}\frac{\partial T}{\partial k}\\ 
    143143  \intertext{and in the k-direction resulting from the lateral tracer gradients} 
    144   \label{eq:i31c} 
     144  \label{eq:TRIADS_i31c} 
    145145  f_{31}+f_{32}=& {A^{lT}} r_1\frac{1}{e_1}\frac{\partial T}{\partial i}+{A^{lT}} r_2\frac{1}{e_1}\frac{\partial T}{\partial i} 
    146146\end{align} 
     
    148148The vertical diffusive flux associated with the $_{33}$ component of the small angle diffusion tensor is 
    149149\begin{equation} 
    150   \label{eq:i33c} 
     150  \label{eq:TRIADS_i33c} 
    151151  f_{33}=-{A^{lT}}(r_1^2 +r_2^2) \frac{1}{e_3}\frac{\partial T}{\partial k}. 
    152152\end{equation} 
     
    157157The following description will describe the fluxes on the $i$-$k$ plane. 
    158158 
    159 There is no natural discretization for the $i$-component of the skew-flux, \autoref{eq:i13c}, 
     159There is no natural discretization for the $i$-component of the skew-flux, \autoref{eq:TRIADS_i13c}, 
    160160as although it must be evaluated at $u$-points, 
    161161it involves vertical gradients (both for the tracer and the slope $r_1$), defined at $w$-points. 
    162 Similarly, the vertical skew flux, \autoref{eq:i31c}, 
     162Similarly, the vertical skew flux, \autoref{eq:TRIADS_i31c}, 
    163163is evaluated at $w$-points but involves horizontal gradients defined at $u$-points. 
    164164 
     
    166166 
    167167The straightforward approach to discretize the lateral skew flux 
    168 \autoref{eq:i13c} from tracer cell $i,k$ to $i+1,k$, introduced in 1995 into OPA, 
    169 \autoref{eq:tra_ldf_iso}, is to calculate a mean vertical gradient at the $u$-point from 
     168\autoref{eq:TRIADS_i13c} from tracer cell $i,k$ to $i+1,k$, introduced in 1995 into OPA, 
     169\autoref{eq:TRA_ldf_iso}, is to calculate a mean vertical gradient at the $u$-point from 
    170170the average of the four surrounding vertical tracer gradients, and multiply this by a mean slope at the $u$-point, 
    171171calculated from the averaged surrounding vertical density gradients. 
    172172The total area-integrated skew-flux (flux per unit area in $ijk$ space) from tracer cell $i,k$ to $i+1,k$, 
    173173noting that the $e_{{3}_{i+1/2}^k}$ in the area $e{_{3}}_{i+1/2}^k{e_{2}}_{i+1/2}i^k$ at the $u$-point cancels out with 
    174 the $1/{e_{3}}_{i+1/2}^k$ associated with the vertical tracer gradient, is then \autoref{eq:tra_ldf_iso} 
     174the $1/{e_{3}}_{i+1/2}^k$ associated with the vertical tracer gradient, is then \autoref{eq:TRA_ldf_iso} 
    175175\[ 
    176176  \left(F_u^{13} \right)_{i+\frac{1}{2}}^k = {A}_{i+\frac{1}{2}}^k 
     
    205205    \includegraphics[width=\textwidth]{Fig_GRIFF_triad_fluxes} 
    206206    \caption{ 
    207       \protect\label{fig:ISO_triad} 
     207      \protect\label{fig:TRIADS_ISO_triad} 
    208208      (a) Arrangement of triads $S_i$ and tracer gradients to 
    209209      give lateral tracer flux from box $i,k$ to $i+1,k$ 
     
    217217the corresponding `triad' slope calculated from the lateral density gradient across the $u$-point divided by 
    218218the vertical density gradient at the same $w$-point as the tracer gradient. 
    219 See \autoref{fig:ISO_triad}a, where the thick lines denote the tracer gradients, 
     219See \autoref{fig:TRIADS_ISO_triad}a, where the thick lines denote the tracer gradients, 
    220220and the thin lines the corresponding triads, with slopes $s_1, \dotsc s_4$. 
    221221The total area-integrated skew-flux from tracer cell $i,k$ to $i+1,k$ 
    222222\begin{multline} 
    223   \label{eq:i13} 
     223  \label{eq:TRIADS_i13} 
    224224  \left( F_u^{13}  \right)_{i+\frac{1}{2}}^k = {A}_{i+1}^k a_1 s_1 
    225225  \delta_{k+\frac{1}{2}} \left[ T^{i+1} 
     
    235235This discretization gives a much closer stencil, and disallows the two-point computational modes. 
    236236 
    237 The vertical skew flux \autoref{eq:i31c} from tracer cell $i,k$ to $i,k+1$ at 
    238 the $w$-point $i,k+\frac{1}{2}$ is constructed similarly (\autoref{fig:ISO_triad}b) by 
     237The vertical skew flux \autoref{eq:TRIADS_i31c} from tracer cell $i,k$ to $i,k+1$ at 
     238the $w$-point $i,k+\frac{1}{2}$ is constructed similarly (\autoref{fig:TRIADS_ISO_triad}b) by 
    239239multiplying lateral tracer gradients from each of the four surrounding $u$-points by the appropriate triad slope: 
    240240\begin{multline} 
    241   \label{eq:i31} 
     241  \label{eq:TRIADS_i31} 
    242242  \left( F_w^{31} \right) _i ^{k+\frac{1}{2}} =  {A}_i^{k+1} a_{1}' 
    243243  s_{1}' \delta_{i-\frac{1}{2}} \left[ T^{k+1} \right]/{e_{3u}}_{i-\frac{1}{2}}^{k+1} 
     
    250250(appearing in both the vertical and lateral gradient), 
    251251and the $u$- and $w$-points $(i+i_p,k)$, $(i,k+k_p)$ at the centres of the `arms' of the triad as follows 
    252 (see also \autoref{fig:ISO_triad}): 
    253 \begin{equation} 
    254   \label{eq:R} 
     252(see also \autoref{fig:TRIADS_ISO_triad}): 
     253\begin{equation} 
     254  \label{eq:TRIADS_R} 
    255255  _i^k \mathbb{R}_{i_p}^{k_p} 
    256256  =-\frac{ {e_{3w}}_{\,i}^{\,k+k_p}} { {e_{1u}}_{\,i+i_p}^{\,k}} 
     
    269269    \includegraphics[width=\textwidth]{Fig_GRIFF_qcells} 
    270270    \caption{ 
    271       \protect\label{fig:qcells} 
     271      \protect\label{fig:TRIADS_qcells} 
    272272      Triad notation for quarter cells. $T$-cells are inside boxes, 
    273273      while the  $i+\fractext{1}{2},k$ $u$-cell is shaded in green and 
     
    278278% >>>>>>>>>>>>>>>>>>>>>>>>>>>> 
    279279 
    280 Each triad $\{_i^{k}\:_{i_p}^{k_p}\}$ is associated (\autoref{fig:qcells}) with the quarter cell that is 
     280Each triad $\{_i^{k}\:_{i_p}^{k_p}\}$ is associated (\autoref{fig:TRIADS_qcells}) with the quarter cell that is 
    281281the intersection of the $i,k$ $T$-cell, the $i+i_p,k$ $u$-cell and the $i,k+k_p$ $w$-cell. 
    282 Expressing the slopes $s_i$ and $s'_i$ in \autoref{eq:i13} and \autoref{eq:i31} in this notation, 
     282Expressing the slopes $s_i$ and $s'_i$ in \autoref{eq:TRIADS_i13} and \autoref{eq:TRIADS_i31} in this notation, 
    283283we have \eg\ \ $s_1=s'_1={\:}_i^k \mathbb{R}_{1/2}^{1/2}$. 
    284284Each triad slope $_i^k\mathbb{R}_{i_p}^{k_p}$ is used once (as an $s$) to 
     
    289289and we notate these areas, similarly to the triad slopes, 
    290290as $_i^k{\mathbb{A}_u}_{i_p}^{k_p}$, $_i^k{\mathbb{A}_w}_{i_p}^{k_p}$, 
    291 where \eg\ in \autoref{eq:i13} $a_{1}={\:}_i^k{\mathbb{A}_u}_{1/2}^{1/2}$, 
    292 and in \autoref{eq:i31} $a'_{1}={\:}_i^k{\mathbb{A}_w}_{1/2}^{1/2}$. 
     291where \eg\ in \autoref{eq:TRIADS_i13} $a_{1}={\:}_i^k{\mathbb{A}_u}_{1/2}^{1/2}$, 
     292and in \autoref{eq:TRIADS_i31} $a'_{1}={\:}_i^k{\mathbb{A}_w}_{1/2}^{1/2}$. 
    293293 
    294294\subsection{Full triad fluxes} 
     
    299299tracer cell $i,k$ to $i+1,k$ coming from the $_{11}$ term of the diffusion tensor takes the form 
    300300\begin{equation} 
    301   \label{eq:i11} 
     301  \label{eq:TRIADS_i11} 
    302302  \left( F_u^{11} \right) _{i+\frac{1}{2}} ^{k} = 
    303303  - \left( {A}_i^{k+1} a_{1} + {A}_i^{k+1} a_{2} + {A}_i^k 
     
    305305  \frac{\delta_{i+1/2} \left[ T^k\right]}{{e_{1u}}_{\,i+1/2}^{\,k}}, 
    306306\end{equation} 
    307 where the areas $a_i$ are as in \autoref{eq:i13}. 
    308 In this case, separating the total lateral flux, the sum of \autoref{eq:i13} and \autoref{eq:i11}, 
     307where the areas $a_i$ are as in \autoref{eq:TRIADS_i13}. 
     308In this case, separating the total lateral flux, the sum of \autoref{eq:TRIADS_i13} and \autoref{eq:TRIADS_i11}, 
    309309into triad components, a lateral tracer flux 
    310310\begin{equation} 
    311   \label{eq:latflux-triad} 
     311  \label{eq:TRIADS_latflux-triad} 
    312312  _i^k {\mathbb{F}_u}_{i_p}^{k_p} (T) = - {A}_i^k{ \:}_i^k{\mathbb{A}_u}_{i_p}^{k_p} 
    313313  \left( 
     
    322322the lateral density flux associated with each triad separately disappears. 
    323323\begin{equation} 
    324   \label{eq:latflux-rho} 
     324  \label{eq:TRIADS_latflux-rho} 
    325325  {\mathbb{F}_u}_{i_p}^{k_p} (\rho)=-\alpha _i^k {\:}_i^k {\mathbb{F}_u}_{i_p}^{k_p} (T) + \beta_i^k {\:}_i^k {\mathbb{F}_u}_{i_p}^{k_p} (S)=0 
    326326\end{equation} 
     
    328328tracer cell $i,k$ to $i+1,k$ must also vanish since it is a sum of four such triad fluxes. 
    329329 
    330 The squared slope $r_1^2$ in the expression \autoref{eq:i33c} for the $_{33}$ component is also expressed in 
     330The squared slope $r_1^2$ in the expression \autoref{eq:TRIADS_i33c} for the $_{33}$ component is also expressed in 
    331331terms of area-weighted squared triad slopes, 
    332332so the area-integrated vertical flux from tracer cell $i,k$ to $i,k+1$ resulting from the $r_1^2$ term is 
    333333\begin{equation} 
    334   \label{eq:i33} 
     334  \label{eq:TRIADS_i33} 
    335335  \left( F_w^{33} \right) _i^{k+\frac{1}{2}} = 
    336336  - \left( {A}_i^{k+1} a_{1}' s_{1}'^2 
     
    339339    + {A}_i^k a_{4}' s_{4}'^2 \right)\delta_{k+\frac{1}{2}} \left[ T^{i+1} \right], 
    340340\end{equation} 
    341 where the areas $a'$ and slopes $s'$ are the same as in \autoref{eq:i31}. 
    342 Then, separating the total vertical flux, the sum of \autoref{eq:i31} and \autoref{eq:i33}, 
     341where the areas $a'$ and slopes $s'$ are the same as in \autoref{eq:TRIADS_i31}. 
     342Then, separating the total vertical flux, the sum of \autoref{eq:TRIADS_i31} and \autoref{eq:TRIADS_i33}, 
    343343into triad components, a vertical flux 
    344344\begin{align} 
    345   \label{eq:vertflux-triad} 
     345  \label{eq:TRIADS_vertflux-triad} 
    346346  _i^k {\mathbb{F}_w}_{i_p}^{k_p} (T) 
    347347  &= {A}_i^k{\: }_i^k{\mathbb{A}_w}_{i_p}^{k_p} 
     
    352352    \right) \\ 
    353353  &= - \left(\left.{ }_i^k{\mathbb{A}_w}_{i_p}^{k_p}\right/{ }_i^k{\mathbb{A}_u}_{i_p}^{k_p}\right) 
    354     {_i^k\mathbb{R}_{i_p}^{k_p}}{\: }_i^k{\mathbb{F}_u}_{i_p}^{k_p} (T) \label{eq:vertflux-triad2} 
     354    {_i^k\mathbb{R}_{i_p}^{k_p}}{\: }_i^k{\mathbb{F}_u}_{i_p}^{k_p} (T) \label{eq:TRIADS_vertflux-triad2} 
    355355\end{align} 
    356356may be associated with each triad. 
     
    361361tracer cell $i,k$ to $i,k+1$ must also vanish since it is a sum of four such triad fluxes. 
    362362 
    363 We can explicitly identify (\autoref{fig:qcells}) the triads associated with the $s_i$, $a_i$, 
    364 and $s'_i$, $a'_i$ used in the definition of the $u$-fluxes and $w$-fluxes in \autoref{eq:i31}, 
    365 \autoref{eq:i13}, \autoref{eq:i11} \autoref{eq:i33} and \autoref{fig:ISO_triad} to write out 
     363We can explicitly identify (\autoref{fig:TRIADS_qcells}) the triads associated with the $s_i$, $a_i$, 
     364and $s'_i$, $a'_i$ used in the definition of the $u$-fluxes and $w$-fluxes in \autoref{eq:TRIADS_i31}, 
     365\autoref{eq:TRIADS_i13}, \autoref{eq:TRIADS_i11} \autoref{eq:TRIADS_i33} and \autoref{fig:TRIADS_ISO_triad} to write out 
    366366the iso-neutral fluxes at $u$- and $w$-points as sums of the triad fluxes that cross the $u$- and $w$-faces: 
    367 %(\autoref{fig:ISO_triad}): 
     367%(\autoref{fig:TRIADS_ISO_triad}): 
    368368\begin{flalign} 
    369   \label{eq:iso_flux} \vect{F}_{\mathrm{iso}}(T) &\equiv 
     369  \label{eq:TRIADS_iso_flux} \vect{F}_{\mathrm{iso}}(T) &\equiv 
    370370  \sum_{\substack{i_p,\,k_p}} 
    371371  \begin{pmatrix} 
     
    376376 
    377377\subsection{Ensuring the scheme does not increase tracer variance} 
    378 \label{subsec:variance} 
     378\label{subsec:TRIADS_variance} 
    379379 
    380380We now require that this operator should not increase the globally-integrated tracer variance. 
     
    400400    &= -T_{i+i_p-1/2}^k{\;} _i^k{\mathbb{F}_u}_{i_p}^{k_p} (T) \quad + \quad  T_{i+i_p+1/2}^k 
    401401    {\;}_i^k{\mathbb{F}_u}_{i_p}^{k_p} (T) \\ 
    402     &={\;} _i^k{\mathbb{F}_u}_{i_p}^{k_p} (T)\,\delta_{i+ i_p}[T^k], \label{eq:dvar_iso_i} 
     402    &={\;} _i^k{\mathbb{F}_u}_{i_p}^{k_p} (T)\,\delta_{i+ i_p}[T^k], \label{eq:TRIADS_dvar_iso_i} 
    403403  \end{aligned} 
    404404\end{multline} 
     
    406406the $T$-points $i,k+k_p-\fractext{1}{2}$ (above) and $i,k+k_p+\fractext{1}{2}$ (below) of 
    407407\begin{equation} 
    408   \label{eq:dvar_iso_k} 
     408  \label{eq:TRIADS_dvar_iso_k} 
    409409  _i^k{\mathbb{F}_w}_{i_p}^{k_p} (T) \,\delta_{k+ k_p}[T^i]. 
    410410\end{equation} 
    411411The total variance tendency driven by the triad is the sum of these two. 
    412412Expanding $_i^k{\mathbb{F}_u}_{i_p}^{k_p} (T)$ and $_i^k{\mathbb{F}_w}_{i_p}^{k_p} (T)$ with 
    413 \autoref{eq:latflux-triad} and \autoref{eq:vertflux-triad}, it is 
     413\autoref{eq:TRIADS_latflux-triad} and \autoref{eq:TRIADS_vertflux-triad}, it is 
    414414\begin{multline*} 
    415415  -{A}_i^k\left \{ 
     
    430430be related to a triad volume $_i^k\mathbb{V}_{i_p}^{k_p}$ by 
    431431\begin{equation} 
    432   \label{eq:V-A} 
     432  \label{eq:TRIADS_V-A} 
    433433  _i^k\mathbb{V}_{i_p}^{k_p} 
    434434  ={\;}_i^k{\mathbb{A}_u}_{i_p}^{k_p}\,{e_{1u}}_{\,i + i_p}^{\,k} 
     
    437437the variance tendency reduces to the perfect square 
    438438\begin{equation} 
    439   \label{eq:perfect-square} 
     439  \label{eq:TRIADS_perfect-square} 
    440440  -{A}_i^k{\:} _i^k\mathbb{V}_{i_p}^{k_p} 
    441441  \left( 
     
    445445  \right)^2\leq 0. 
    446446\end{equation} 
    447 Thus, the constraint \autoref{eq:V-A} ensures that the fluxes 
    448 (\autoref{eq:latflux-triad}, \autoref{eq:vertflux-triad}) associated with 
     447Thus, the constraint \autoref{eq:TRIADS_V-A} ensures that the fluxes 
     448(\autoref{eq:TRIADS_latflux-triad}, \autoref{eq:TRIADS_vertflux-triad}) associated with 
    449449a given slope triad $_i^k\mathbb{R}_{i_p}^{k_p}$ do not increase the net variance. 
    450450Since the total fluxes are sums of such fluxes from the various triads, this constraint, applied to all triads, 
    451451is sufficient to ensure that the globally integrated variance does not increase. 
    452452 
    453 The expression \autoref{eq:V-A} can be interpreted as a discretization of the global integral 
    454 \begin{equation} 
    455   \label{eq:cts-var} 
     453The expression \autoref{eq:TRIADS_V-A} can be interpreted as a discretization of the global integral 
     454\begin{equation} 
     455  \label{eq:TRIADS_cts-var} 
    456456  \frac{\partial}{\partial t}\int\!\fractext{1}{2} T^2\, dV = 
    457457  \int\!\mathbf{F}\cdot\nabla T\, dV, 
     
    477477\citet{griffies.gnanadesikan.ea_JPO98} identifies these $_i^k\mathbb{V}_{i_p}^{k_p}$ as the volumes of the quarter cells, 
    478478defined in terms of the distances between $T$, $u$,$f$ and $w$-points. 
    479 This is the natural discretization of \autoref{eq:cts-var}. 
     479This is the natural discretization of \autoref{eq:TRIADS_cts-var}. 
    480480The \NEMO\ model, however, operates with scale factors instead of grid sizes, 
    481481and scale factors for the quarter cells are not defined. 
    482482Instead, therefore we simply choose 
    483483\begin{equation} 
    484   \label{eq:V-NEMO} 
     484  \label{eq:TRIADS_V-NEMO} 
    485485  _i^k\mathbb{V}_{i_p}^{k_p}=\fractext{1}{4} {b_u}_{i+i_p}^k, 
    486486\end{equation} 
     
    489489the lateral flux from tracer cell $i,k$ to $i+1,k$ reduces to the classical form 
    490490\begin{equation} 
    491   \label{eq:lat-normal} 
     491  \label{eq:TRIADS_lat-normal} 
    492492  -\overline{A}_{\,i+1/2}^k\; 
    493493  \frac{{b_u}_{i+1/2}^k}{{e_{1u}}_{\,i + i_p}^{\,k}} 
     
    497497In fact if the diffusive coefficient is defined at $u$-points, 
    498498so that we employ ${A}_{i+i_p}^k$ instead of  ${A}_i^k$ in the definitions of the triad fluxes 
    499 \autoref{eq:latflux-triad} and \autoref{eq:vertflux-triad}, 
     499\autoref{eq:TRIADS_latflux-triad} and \autoref{eq:TRIADS_vertflux-triad}, 
    500500we can replace $\overline{A}_{\,i+1/2}^k$ by $A_{i+1/2}^k$ in the above. 
    501501 
     
    503503 
    504504The iso-neutral fluxes at $u$- and $w$-points are the sums of the triad fluxes that 
    505 cross the $u$- and $w$-faces \autoref{eq:iso_flux}: 
     505cross the $u$- and $w$-faces \autoref{eq:TRIADS_iso_flux}: 
    506506\begin{subequations} 
    507   % \label{eq:alltriadflux} 
     507  % \label{eq:TRIADS_alltriadflux} 
    508508  \begin{flalign*} 
    509     % \label{eq:vect_isoflux} 
     509    % \label{eq:TRIADS_vect_isoflux} 
    510510    \vect{F}_{\mathrm{iso}}(T) &\equiv 
    511511    \sum_{\substack{i_p,\,k_p}} 
     
    515515    \end{pmatrix}, 
    516516  \end{flalign*} 
    517   where \autoref{eq:latflux-triad}: 
     517  where \autoref{eq:TRIADS_latflux-triad}: 
    518518  \begin{align} 
    519     \label{eq:triadfluxu} 
     519    \label{eq:TRIADS_triadfluxu} 
    520520    _i^k {\mathbb{F}_u}_{i_p}^{k_p} (T) &= - {A}_i^k{ 
    521521                                          \:}\frac{{{}_i^k\mathbb{V}}_{i_p}^{k_p}}{{e_{1u}}_{\,i + i_p}^{\,k}} 
     
    532532                                          -\ \left({_i^k\mathbb{R}_{i_p}^{k_p}}\right)^2 \ 
    533533                                          \frac{ \delta_{k+k_p} [T^i] }{{e_{3w}}_{\,i}^{\,k+k_p} } 
    534                                           \right),\label{eq:triadfluxw} 
     534                                          \right),\label{eq:TRIADS_triadfluxw} 
    535535  \end{align} 
    536   with \autoref{eq:V-NEMO} 
     536  with \autoref{eq:TRIADS_V-NEMO} 
    537537  \[ 
    538     % \label{eq:V-NEMO2} 
     538    % \label{eq:TRIADS_V-NEMO2} 
    539539    _i^k{\mathbb{V}}_{i_p}^{k_p}=\fractext{1}{4} {b_u}_{i+i_p}^k. 
    540540  \] 
    541541\end{subequations} 
    542542 
    543 The divergence of the expression \autoref{eq:iso_flux} for the fluxes gives the iso-neutral diffusion tendency at 
     543The divergence of the expression \autoref{eq:TRIADS_iso_flux} for the fluxes gives the iso-neutral diffusion tendency at 
    544544each tracer point: 
    545545\[ 
    546   % \label{eq:iso_operator} 
     546  % \label{eq:TRIADS_iso_operator} 
    547547  D_l^T = \frac{1}{b_T} 
    548548  \sum_{\substack{i_p,\,k_p}} \left\{ \delta_{i} \left[{_{i+1/2-i_p}^k 
     
    555555\item[$\bullet$ horizontal diffusion] 
    556556  The discretization of the diffusion operator recovers the traditional five-point Laplacian 
    557   \autoref{eq:lat-normal} in the limit of flat iso-neutral direction: 
     557  \autoref{eq:TRIADS_lat-normal} in the limit of flat iso-neutral direction: 
    558558  \[ 
    559     % \label{eq:iso_property0} 
     559    % \label{eq:TRIADS_iso_property0} 
    560560    D_l^T = \frac{1}{b_T} \ 
    561561    \delta_{i} \left[ \frac{e_{2u}\,e_{3u}}{e_{1u}} \; 
     
    565565 
    566566\item[$\bullet$ implicit treatment in the vertical] 
    567   Only tracer values associated with a single water column appear in the expression \autoref{eq:i33} for 
     567  Only tracer values associated with a single water column appear in the expression \autoref{eq:TRIADS_i33} for 
    568568  the $_{33}$ fluxes, vertical fluxes driven by vertical gradients. 
    569569  This is of paramount importance since it means that a time-implicit algorithm can be used to 
     
    582582\item[$\bullet$ pure iso-neutral operator] 
    583583  The iso-neutral flux of locally referenced potential density is zero. 
    584   See \autoref{eq:latflux-rho} and \autoref{eq:vertflux-triad2}. 
     584  See \autoref{eq:TRIADS_latflux-rho} and \autoref{eq:TRIADS_vertflux-triad2}. 
    585585 
    586586\item[$\bullet$ conservation of tracer] 
    587587  The iso-neutral diffusion conserves tracer content, \ie 
    588588  \[ 
    589     % \label{eq:iso_property1} 
     589    % \label{eq:TRIADS_iso_property1} 
    590590    \sum_{i,j,k} \left\{ D_l^T \      b_T \right\} = 0 
    591591  \] 
     
    595595  The iso-neutral diffusion does not increase the tracer variance, \ie 
    596596  \[ 
    597     % \label{eq:iso_property2} 
     597    % \label{eq:TRIADS_iso_property2} 
    598598    \sum_{i,j,k} \left\{ T \ D_l^T      \ b_T \right\} \leq 0 
    599599  \] 
    600   The property is demonstrated in \autoref{subsec:variance} above. 
     600  The property is demonstrated in \autoref{subsec:TRIADS_variance} above. 
    601601  It is a key property for a diffusion term. 
    602602  It means that it is also a dissipation term, 
     
    608608  The iso-neutral diffusion operator is self-adjoint, \ie 
    609609  \begin{equation} 
    610     \label{eq:iso_property3} 
     610    \label{eq:TRIADS_iso_property3} 
    611611    \sum_{i,j,k} \left\{ S \ D_l^T \ b_T \right\} = \sum_{i,j,k} \left\{ D_l^S \ T \ b_T \right\} 
    612612  \end{equation} 
     
    614614  We just have to apply the same routine. 
    615615  This property can be demonstrated similarly to the proof of the `no increase of tracer variance' property. 
    616   The contribution by a single triad towards the left hand side of \autoref{eq:iso_property3}, 
    617   can be found by replacing $\delta[T]$ by $\delta[S]$ in \autoref{eq:dvar_iso_i} and \autoref{eq:dvar_iso_k}. 
    618   This results in a term similar to \autoref{eq:perfect-square}, 
     616  The contribution by a single triad towards the left hand side of \autoref{eq:TRIADS_iso_property3}, 
     617  can be found by replacing $\delta[T]$ by $\delta[S]$ in \autoref{eq:TRIADS_dvar_iso_i} and \autoref{eq:TRIADS_dvar_iso_k}. 
     618  This results in a term similar to \autoref{eq:TRIADS_perfect-square}, 
    619619  \[ 
    620     % \label{eq:TScovar} 
     620    % \label{eq:TRIADS_TScovar} 
    621621    - {A}_i^k{\:} _i^k\mathbb{V}_{i_p}^{k_p} 
    622622    \left( 
     
    632632  \] 
    633633This is symmetrical in $T $ and $S$, so exactly the same term arises from 
    634 the discretization of this triad's contribution towards the RHS of \autoref{eq:iso_property3}. 
     634the discretization of this triad's contribution towards the RHS of \autoref{eq:TRIADS_iso_property3}. 
    635635\end{description} 
    636636 
    637637\subsection{Treatment of the triads at the boundaries} 
    638 \label{sec:iso_bdry} 
     638\label{sec:TRIADS_iso_bdry} 
    639639 
    640640The triad slope can only be defined where both the grid boxes centred at the end of the arms exist. 
    641641Triads that would poke up through the upper ocean surface into the atmosphere, 
    642642or down into the ocean floor, must be masked out. 
    643 See \autoref{fig:bdry_triads}. 
     643See \autoref{fig:TRIADS_bdry_triads}. 
    644644Surface layer triads \triad{i}{1}{R}{1/2}{-1/2} (magenta) and \triad{i+1}{1}{R}{-1/2}{-1/2} (blue) that 
    645 require density to be specified above the ocean surface are masked (\autoref{fig:bdry_triads}a): 
     645require density to be specified above the ocean surface are masked (\autoref{fig:TRIADS_bdry_triads}a): 
    646646this ensures that lateral tracer gradients produce no flux through the ocean surface. 
    647647However, to prevent surface noise, it is customary to retain the $_{11}$ contributions towards 
    648648the lateral triad fluxes \triad[u]{i}{1}{F}{1/2}{-1/2} and \triad[u]{i+1}{1}{F}{-1/2}{-1/2}; 
    649649this drives diapycnal tracer fluxes. 
    650 Similar comments apply to triads that would intersect the ocean floor (\autoref{fig:bdry_triads}b). 
     650Similar comments apply to triads that would intersect the ocean floor (\autoref{fig:TRIADS_bdry_triads}b). 
    651651Note that both near bottom triad slopes \triad{i}{k}{R}{1/2}{1/2} and \triad{i+1}{k}{R}{-1/2}{1/2} are masked when 
    652652either of the $i,k+1$ or $i+1,k+1$ tracer points is masked, \ie\ the $i,k+1$ $u$-point is masked. 
     
    662662    \includegraphics[width=\textwidth]{Fig_GRIFF_bdry_triads} 
    663663    \caption{ 
    664       \protect\label{fig:bdry_triads} 
     664      \protect\label{fig:TRIADS_bdry_triads} 
    665665      (a) Uppermost model layer $k=1$ with $i,1$ and $i+1,1$ tracer points (black dots), 
    666666      and $i+1/2,1$ $u$-point (blue square). 
     
    683683 
    684684\subsection{ Limiting of the slopes within the interior} 
    685 \label{sec:limit} 
     685\label{sec:TRIADS_limit} 
    686686 
    687687As discussed in \autoref{subsec:LDF_slp_iso}, 
     
    693693It is of course relevant to the iso-neutral slopes $\tilde{r}_i=r_i+\sigma_i$ relative to geopotentials 
    694694(here the $\sigma_i$ are the slopes of the coordinate surfaces relative to geopotentials) 
    695 \autoref{eq:PE_slopes_eiv} rather than the slope $r_i$ relative to coordinate surfaces, so we require 
     695\autoref{eq:MB_slopes_eiv} rather than the slope $r_i$ relative to coordinate surfaces, so we require 
    696696\[ 
    697697  |\tilde{r}_i|\leq \tilde{r}_\mathrm{max}=0.01. 
     
    700700Each individual triad slope 
    701701\begin{equation} 
    702   \label{eq:Rtilde} 
     702  \label{eq:TRIADS_Rtilde} 
    703703  _i^k\tilde{\mathbb{R}}_{i_p}^{k_p} = {}_i^k\mathbb{R}_{i_p}^{k_p}  + \frac{\delta_{i+i_p}[z_T^k]}{{e_{1u}}_{\,i + i_p}^{\,k}} 
    704704\end{equation} 
     
    711711 
    712712\subsection{Tapering within the surface mixed layer} 
    713 \label{sec:taper} 
     713\label{sec:TRIADS_taper} 
    714714 
    715715Additional tapering of the iso-neutral fluxes is necessary within the surface mixed layer. 
     
    717717 
    718718\subsubsection{Linear slope tapering within the surface mixed layer} 
    719 \label{sec:lintaper} 
     719\label{sec:TRIADS_lintaper} 
    720720 
    721721This is the option activated by the default choice \np{ln\_triad\_iso}\forcode{ = .false.}. 
    722722Slopes $\tilde{r}_i$ relative to geopotentials are tapered linearly from their value immediately below 
    723 the mixed layer to zero at the surface, as described in option (c) of \autoref{fig:eiv_slp}, to values 
    724 \begin{equation} 
    725   \label{eq:rmtilde} 
     723the mixed layer to zero at the surface, as described in option (c) of \autoref{fig:LDF_eiv_slp}, to values 
     724\begin{equation} 
     725  \label{eq:TRIADS_rmtilde} 
    726726  \rMLt = -\frac{z}{h}\left.\tilde{r}_i\right|_{z=-h}\quad \text{ for  } z>-h, 
    727727\end{equation} 
    728728and then the $r_i$ relative to vertical coordinate surfaces are appropriately adjusted to 
    729729\[ 
    730   % \label{eq:rm} 
     730  % \label{eq:TRIADS_rm} 
    731731  \rML =\rMLt -\sigma_i \quad \text{ for  } z>-h. 
    732732\] 
    733733Thus the diffusion operator within the mixed layer is given by: 
    734734\[ 
    735   % \label{eq:iso_tensor_ML} 
     735  % \label{eq:TRIADS_iso_tensor_ML} 
    736736  D^{lT}=\nabla {\mathrm {\mathbf .}}\left( {A^{lT}\;\Re \;\nabla T} \right) \qquad 
    737737  \mbox{with}\quad \;\;\Re =\left( {{ 
     
    747747in isopycnal layers immediately below, in the thermocline. 
    748748It is consistent with the way the $\tilde{r}_i$ are tapered within the mixed layer 
    749 (see \autoref{sec:taperskew} below) so as to ensure a uniform GM eddy-induced velocity throughout the mixed layer. 
     749(see \autoref{sec:TRIADS_taperskew} below) so as to ensure a uniform GM eddy-induced velocity throughout the mixed layer. 
    750750However, it gives a downwards density flux and so acts so as to reduce potential energy in the same way as 
    751 does the slope limiting discussed above in \autoref{sec:limit}. 
    752   
    753 As in \autoref{sec:limit} above, the tapering \autoref{eq:rmtilde} is applied separately to 
     751does the slope limiting discussed above in \autoref{sec:TRIADS_limit}. 
     752 
     753As in \autoref{sec:TRIADS_limit} above, the tapering \autoref{eq:TRIADS_rmtilde} is applied separately to 
    754754each triad $_i^k\tilde{\mathbb{R}}_{i_p}^{k_p}$, and the $_i^k\mathbb{R}_{i_p}^{k_p}$ adjusted. 
    755755For clarity, we assume $z$-coordinates in the following; 
    756756the conversion from $\mathbb{R}$ to $\tilde{\mathbb{R}}$ and back to $\mathbb{R}$ follows exactly as 
    757 described above by \autoref{eq:Rtilde}. 
     757described above by \autoref{eq:TRIADS_Rtilde}. 
    758758\begin{enumerate} 
    759759\item 
    760760  Mixed-layer depth is defined so as to avoid including regions of weak vertical stratification in 
    761761  the slope definition. 
    762   At each $i,j$ (simplified to $i$ in \autoref{fig:MLB_triad}), 
     762  At each $i,j$ (simplified to $i$ in \autoref{fig:TRIADS_MLB_triad}), 
    763763  we define the mixed-layer by setting the vertical index of the tracer point immediately below the mixed layer, 
    764764  $k_{\mathrm{ML}}$, as the maximum $k$ (shallowest tracer point) such that 
    765765  the potential density ${\rho_0}_{i,k}>{\rho_0}_{i,k_{10}}+\Delta\rho_c$, 
    766766  where $i,k_{10}$ is the tracer gridbox within which the depth reaches 10~m. 
    767   See the left side of \autoref{fig:MLB_triad}. 
     767  See the left side of \autoref{fig:TRIADS_MLB_triad}. 
    768768  We use the $k_{10}$-gridbox instead of the surface gridbox to avoid problems \eg\ with thin daytime mixed-layers. 
    769769  Currently we use the same $\Delta\rho_c=0.01\;\mathrm{kg\:m^{-3}}$ for ML triad tapering as is used to 
     
    776776  This is to ensure that the vertical density gradients associated with 
    777777  these basal triad slopes ${\:}_i{\mathbb{R}_{\mathrm{base}}}_{\,i_p}^{k_p}$ are representative of the thermocline. 
    778   The four basal triads defined in the bottom part of \autoref{fig:MLB_triad} are then 
     778  The four basal triads defined in the bottom part of \autoref{fig:TRIADS_MLB_triad} are then 
    779779  \begin{align*} 
    780780    {\:}_i{\mathbb{R}_{\mathrm{base}}}_{\,i_p}^{k_p} &= 
    781781                                                       {\:}^{k_{\mathrm{ML}}-k_p-1/2}_i{\mathbb{R}_{\mathrm{base}}}_{\,i_p}^{k_p}, 
    782                                                        % \label{eq:Rbase} 
     782                                                       % \label{eq:TRIADS_Rbase} 
    783783    \\ 
    784784    \intertext{with \eg\ the green triad} 
     
    789789the $w$-point $i,k_{\mathrm{ML}}-1/2$ lying \emph{below} the $i,k_{\mathrm{ML}}$ tracer point, so it is this depth 
    790790\[ 
    791   % \label{eq:zbase} 
     791  % \label{eq:TRIADS_zbase} 
    792792  {z_\mathrm{base}}_{\,i}={z_{w}}_{k_\mathrm{ML}-1/2} 
    793793\] 
    794794one gridbox deeper than the diagnosed ML depth $z_{\mathrm{ML}})$ that sets the $h$ used to taper the slopes in 
    795 \autoref{eq:rmtilde}. 
     795\autoref{eq:TRIADS_rmtilde}. 
    796796\item 
    797797  Finally, we calculate the adjusted triads ${\:}_i^k{\mathbb{R}_{\mathrm{ML}}}_{\,i_p}^{k_p}$ within 
     
    805805    {\:}_i^k{\mathbb{R}_{\mathrm{ML}}}_{\,i_p}^{k_p} &= 
    806806                                                       \frac{{z_w}_{k+k_p}}{{z_{\mathrm{base}}}_{\,i}}{\:}_i{\mathbb{R}_{\mathrm{base}}}_{\,i_p}^{k_p}. 
    807                                                        % \label{eq:RML} 
     807                                                       % \label{eq:TRIADS_RML} 
    808808  \end{align*} 
    809809\end{enumerate} 
     
    813813%  \fcapside { 
    814814  \caption{ 
    815     \protect\label{fig:MLB_triad} 
     815    \protect\label{fig:TRIADS_MLB_triad} 
    816816    Definition of mixed-layer depth and calculation of linearly tapered triads. 
    817817    The figure shows a water column at a given $i,j$ (simplified to $i$), with the ocean surface at the top. 
     
    836836 
    837837\subsubsection{Additional truncation of skew iso-neutral flux components} 
    838 \label{subsec:Gerdes-taper} 
     838\label{subsec:TRIADS_Gerdes-taper} 
    839839 
    840840The alternative option is activated by setting \np{ln\_triad\_iso} = true. 
     
    843843but replaces the $\rML$ in the skew term by 
    844844\begin{equation} 
    845   \label{eq:rm*} 
     845  \label{eq:TRIADS_rm*} 
    846846  \rML^*=\left.\rMLt^2\right/\tilde{r}_i-\sigma_i, 
    847847\end{equation} 
    848848giving a ML diffusive operator 
    849849\[ 
    850   % \label{eq:iso_tensor_ML2} 
     850  % \label{eq:TRIADS_iso_tensor_ML2} 
    851851  D^{lT}=\nabla {\mathrm {\mathbf .}}\left( {A^{lT}\;\Re \;\nabla T} \right) \qquad 
    852852  \mbox{with}\quad \;\;\Re =\left( {{ 
     
    881881% ================================================================ 
    882882\section{Eddy induced advection formulated as a skew flux} 
    883 \label{sec:skew-flux} 
     883\label{sec:TRIADS_skew-flux} 
    884884 
    885885\subsection{Continuous skew flux formulation} 
    886 \label{sec:continuous-skew-flux} 
     886\label{sec:TRIADS_continuous-skew-flux} 
    887887 
    888888When Gent and McWilliams's [1990] diffusion is used, an additional advection term is added. 
     
    890890the formulation of which depends on the slopes of iso-neutral surfaces. 
    891891Contrary to the case of iso-neutral mixing, the slopes used here are referenced to the geopotential surfaces, 
    892 \ie\ \autoref{eq:ldfslp_geo} is used in $z$-coordinate, 
    893 and the sum \autoref{eq:ldfslp_geo} + \autoref{eq:ldfslp_iso} in $z^*$ or $s$-coordinates. 
     892\ie\ \autoref{eq:LDF_slp_geo} is used in $z$-coordinate, 
     893and the sum \autoref{eq:LDF_slp_geo} + \autoref{eq:LDF_slp_iso} in $z^*$ or $s$-coordinates. 
    894894 
    895895The eddy induced velocity is given by: 
    896896\begin{subequations} 
    897   % \label{eq:eiv} 
     897  % \label{eq:TRIADS_eiv} 
    898898  \begin{equation} 
    899     \label{eq:eiv_v} 
     899    \label{eq:TRIADS_eiv_v} 
    900900    \begin{split} 
    901901      u^* & = - \frac{1}{e_{3}}\;          \partial_i\psi_1,  \\ 
     
    907907  where the streamfunctions $\psi_i$ are given by 
    908908  \begin{equation} 
    909     \label{eq:eiv_psi} 
     909    \label{eq:TRIADS_eiv_psi} 
    910910    \begin{split} 
    911911      \psi_1 & = A_{e} \; \tilde{r}_1,   \\ 
     
    958958we end up with the skew form of the eddy induced advective fluxes per unit area in $ijk$ space: 
    959959\begin{equation} 
    960   \label{eq:eiv_skew_ijk} 
     960  \label{eq:TRIADS_eiv_skew_ijk} 
    961961  \textbf{F}_\mathrm{eiv}^T = 
    962962  \begin{pmatrix} 
     
    967967The total fluxes per unit physical area are then 
    968968\begin{equation} 
    969   \label{eq:eiv_skew_physical} 
     969  \label{eq:TRIADS_eiv_skew_physical} 
    970970  \begin{split} 
    971971    f^*_1 & = \frac{1}{e_{3}}\; \psi_1 \partial_k T   \\ 
     
    974974\end{split} 
    975975\end{equation} 
    976 Note that \autoref{eq:eiv_skew_physical} takes the same form whatever the vertical coordinate, 
    977 though of course the slopes $\tilde{r}_i$ which define the $\psi_i$ in \autoref{eq:eiv_psi} are relative to 
     976Note that \autoref{eq:TRIADS_eiv_skew_physical} takes the same form whatever the vertical coordinate, 
     977though of course the slopes $\tilde{r}_i$ which define the $\psi_i$ in \autoref{eq:TRIADS_eiv_psi} are relative to 
    978978geopotentials. 
    979979The tendency associated with eddy induced velocity is then simply the convergence of the fluxes 
    980 (\autoref{eq:eiv_skew_ijk}, \autoref{eq:eiv_skew_physical}), so 
    981 \[ 
    982   % \label{eq:skew_eiv_conv} 
     980(\autoref{eq:TRIADS_eiv_skew_ijk}, \autoref{eq:TRIADS_eiv_skew_physical}), so 
     981\[ 
     982  % \label{eq:TRIADS_skew_eiv_conv} 
    983983  \frac{\partial T}{\partial t}= -\frac{1}{e_1 \, e_2 \, e_3 }      \left[ 
    984984    \frac{\partial}{\partial i} \left( e_2 \psi_1 \partial_k T\right) 
     
    993993\subsection{Discrete skew flux formulation} 
    994994 
    995 The skew fluxes in (\autoref{eq:eiv_skew_physical}, \autoref{eq:eiv_skew_ijk}), 
    996 like the off-diagonal terms (\autoref{eq:i13c}, \autoref{eq:i31c}) of the small angle diffusion tensor, 
    997 are best expressed in terms of the triad slopes, as in \autoref{fig:ISO_triad} and 
    998 (\autoref{eq:i13}, \autoref{eq:i31}); 
     995The skew fluxes in (\autoref{eq:TRIADS_eiv_skew_physical}, \autoref{eq:TRIADS_eiv_skew_ijk}), 
     996like the off-diagonal terms (\autoref{eq:TRIADS_i13c}, \autoref{eq:TRIADS_i31c}) of the small angle diffusion tensor, 
     997are best expressed in terms of the triad slopes, as in \autoref{fig:TRIADS_ISO_triad} and 
     998(\autoref{eq:TRIADS_i13}, \autoref{eq:TRIADS_i31}); 
    999999but now in terms of the triad slopes $\tilde{\mathbb{R}}$ relative to geopotentials instead of 
    10001000the $\mathbb{R}$ relative to coordinate surfaces. 
    1001 The discrete form of \autoref{eq:eiv_skew_ijk} using the slopes \autoref{eq:R} and 
     1001The discrete form of \autoref{eq:TRIADS_eiv_skew_ijk} using the slopes \autoref{eq:TRIADS_R} and 
    10021002defining $A_e$ at $T$-points is then given by: 
    10031003 
    10041004\begin{subequations} 
    1005   % \label{eq:allskewflux} 
     1005  % \label{eq:TRIADS_allskewflux} 
    10061006  \begin{flalign*} 
    1007     % \label{eq:vect_skew_flux} 
     1007    % \label{eq:TRIADS_vect_skew_flux} 
    10081008    \vect{F}_{\mathrm{eiv}}(T) &\equiv    \sum_{\substack{i_p,\,k_p}} 
    10091009    \begin{pmatrix} 
     
    10121012    \end{pmatrix}, 
    10131013  \end{flalign*} 
    1014   where the skew flux in the $i$-direction associated with a given triad is (\autoref{eq:latflux-triad}, 
    1015   \autoref{eq:triadfluxu}): 
     1014  where the skew flux in the $i$-direction associated with a given triad is (\autoref{eq:TRIADS_latflux-triad}, 
     1015  \autoref{eq:TRIADS_triadfluxu}): 
    10161016  \begin{align} 
    1017     \label{eq:skewfluxu} 
     1017    \label{eq:TRIADS_skewfluxu} 
    10181018    _i^k {\mathbb{S}_u}_{i_p}^{k_p} (T) &= + \fractext{1}{4} {A_e}_i^k{ 
    10191019                                          \:}\frac{{b_u}_{i+i_p}^k}{{e_{1u}}_{\,i + i_p}^{\,k}} 
     
    10211021                                          \frac{ \delta_{k+k_p} [T^i] }{{e_{3w}}_{\,i}^{\,k+k_p} }, \\ 
    10221022    \intertext{ 
    1023     and \autoref{eq:triadfluxw} in the $k$-direction, changing the sign 
    1024     to be consistent with \autoref{eq:eiv_skew_ijk}: 
     1023    and \autoref{eq:TRIADS_triadfluxw} in the $k$-direction, changing the sign 
     1024    to be consistent with \autoref{eq:TRIADS_eiv_skew_ijk}: 
    10251025    } 
    10261026    _i^k {\mathbb{S}_w}_{i_p}^{k_p} (T) 
    10271027                                        &= -\fractext{1}{4} {A_e}_i^k{\: }\frac{{b_u}_{i+i_p}^k}{{e_{3w}}_{\,i}^{\,k+k_p}} 
    1028                                           {_i^k\tilde{\mathbb{R}}_{i_p}^{k_p}}\frac{ \delta_{i+ i_p}[T^k] }{ {e_{1u}}_{\,i + i_p}^{\,k} }.\label{eq:skewfluxw} 
     1028                                          {_i^k\tilde{\mathbb{R}}_{i_p}^{k_p}}\frac{ \delta_{i+ i_p}[T^k] }{ {e_{1u}}_{\,i + i_p}^{\,k} }.\label{eq:TRIADS_skewfluxw} 
    10291029  \end{align} 
    10301030\end{subequations} 
     
    10381038This can be seen %either from Appendix \autoref{apdx:eiv_skew} or 
    10391039by considering the fluxes associated with a given triad slope $_i^k{\mathbb{R}}_{i_p}^{k_p} (T)$. 
    1040 For, following \autoref{subsec:variance} and \autoref{eq:dvar_iso_i}, 
     1040For, following \autoref{subsec:TRIADS_variance} and \autoref{eq:TRIADS_dvar_iso_i}, 
    10411041the associated horizontal skew-flux $_i^k{\mathbb{S}_u}_{i_p}^{k_p} (T)$ drives a net rate of change of variance, 
    10421042summed over the two $T$-points $i+i_p-\fractext{1}{2},k$ and $i+i_p+\fractext{1}{2},k$, of 
    10431043\begin{equation} 
    1044   \label{eq:dvar_eiv_i} 
     1044  \label{eq:TRIADS_dvar_eiv_i} 
    10451045  _i^k{\mathbb{S}_u}_{i_p}^{k_p} (T)\,\delta_{i+ i_p}[T^k], 
    10461046\end{equation} 
     
    10481048the $T$-points $i,k+k_p-\fractext{1}{2}$ (above) and $i,k+k_p+\fractext{1}{2}$ (below) of 
    10491049\begin{equation} 
    1050   \label{eq:dvar_eiv_k} 
     1050  \label{eq:TRIADS_dvar_eiv_k} 
    10511051  _i^k{\mathbb{S}_w}_{i_p}^{k_p} (T) \,\delta_{k+ k_p}[T^i]. 
    10521052\end{equation} 
    1053 Inspection of the definitions (\autoref{eq:skewfluxu}, \autoref{eq:skewfluxw}) shows that 
    1054 these two variance changes (\autoref{eq:dvar_eiv_i}, \autoref{eq:dvar_eiv_k}) sum to zero. 
     1053Inspection of the definitions (\autoref{eq:TRIADS_skewfluxu}, \autoref{eq:TRIADS_skewfluxw}) shows that 
     1054these two variance changes (\autoref{eq:TRIADS_dvar_eiv_i}, \autoref{eq:TRIADS_dvar_eiv_k}) sum to zero. 
    10551055Hence the two fluxes associated with each triad make no net contribution to the variance budget. 
    10561056 
     
    10641064For the change in gravitational PE driven by the $k$-flux is 
    10651065\begin{align} 
    1066   \label{eq:vert_densityPE} 
     1066  \label{eq:TRIADS_vert_densityPE} 
    10671067  g {e_{3w}}_{\,i}^{\,k+k_p}{\mathbb{S}_w}_{i_p}^{k_p} (\rho) 
    10681068  &=g {e_{3w}}_{\,i}^{\,k+k_p}\left[-\alpha _i^k {\:}_i^k 
    10691069    {\mathbb{S}_w}_{i_p}^{k_p} (T) + \beta_i^k {\:}_i^k 
    10701070    {\mathbb{S}_w}_{i_p}^{k_p} (S) \right]. \notag \\ 
    1071   \intertext{Substituting  ${\:}_i^k {\mathbb{S}_w}_{i_p}^{k_p}$ from \autoref{eq:skewfluxw}, gives} 
     1071  \intertext{Substituting  ${\:}_i^k {\mathbb{S}_w}_{i_p}^{k_p}$ from \autoref{eq:TRIADS_skewfluxw}, gives} 
    10721072  % and separating out 
    10731073  % $\rtriadt{R}=\rtriad{R} + \delta_{i+i_p}[z_T^k]$, 
     
    10801080    \frac{-\alpha_i^k \delta_{k+ k_p}[T^i]+ \beta_i^k\delta_{k+ k_p}[S^i]} {{e_{3w}}_{\,i}^{\,k+k_p}}, 
    10811081\end{align} 
    1082 using the definition of the triad slope $\rtriad{R}$, \autoref{eq:R} to 
     1082using the definition of the triad slope $\rtriad{R}$, \autoref{eq:TRIADS_R} to 
    10831083express $-\alpha _i^k\delta_{i+ i_p}[T^k]+\beta_i^k\delta_{i+ i_p}[S^k]$ in terms of 
    10841084$-\alpha_i^k \delta_{k+ k_p}[T^i]+ \beta_i^k\delta_{k+ k_p}[S^i]$. 
     
    10861086Where the coordinates slope, the $i$-flux gives a PE change 
    10871087\begin{multline} 
    1088   \label{eq:lat_densityPE} 
     1088  \label{eq:TRIADS_lat_densityPE} 
    10891089  g \delta_{i+i_p}[z_T^k] 
    10901090  \left[ 
     
    10961096  \frac{-\alpha_i^k \delta_{k+ k_p}[T^i]+ \beta_i^k\delta_{k+ k_p}[S^i]} {{e_{3w}}_{\,i}^{\,k+k_p}}, 
    10971097\end{multline} 
    1098 (using \autoref{eq:skewfluxu}) and so the total PE change \autoref{eq:vert_densityPE} + 
    1099 \autoref{eq:lat_densityPE} associated with the triad fluxes is 
     1098(using \autoref{eq:TRIADS_skewfluxu}) and so the total PE change \autoref{eq:TRIADS_vert_densityPE} + 
     1099\autoref{eq:TRIADS_lat_densityPE} associated with the triad fluxes is 
    11001100\begin{multline*} 
    1101   % \label{eq:tot_densityPE} 
     1101  % \label{eq:TRIADS_tot_densityPE} 
    11021102  g{e_{3w}}_{\,i}^{\,k+k_p}{\mathbb{S}_w}_{i_p}^{k_p} (\rho) + 
    11031103  g\delta_{i+i_p}[z_T^k] {\:}_i^k {\mathbb{S}_u}_{i_p}^{k_p} (\rho) \\ 
     
    11101110 
    11111111\subsection{Treatment of the triads at the boundaries} 
    1112 \label{sec:skew_bdry} 
    1113  
    1114 Triad slopes \rtriadt{R} used for the calculation of the eddy-induced skew-fluxes are masked at the boundaries  
    1115 in exactly the same way as are the triad slopes \rtriad{R} used for the iso-neutral diffusive fluxes,  
    1116 as described in \autoref{sec:iso_bdry} and \autoref{fig:bdry_triads}.  
    1117 Thus surface layer triads $\triadt{i}{1}{R}{1/2}{-1/2}$ and $\triadt{i+1}{1}{R}{-1/2}{-1/2}$ are masked,  
    1118 and both near bottom triad slopes $\triadt{i}{k}{R}{1/2}{1/2}$ and $\triadt{i+1}{k}{R}{-1/2}{1/2}$ are masked when  
    1119 either of the $i,k+1$ or $i+1,k+1$ tracer points is masked, \ie\ the $i,k+1$ $u$-point is masked.  
     1112\label{sec:TRIADS_skew_bdry} 
     1113 
     1114Triad slopes \rtriadt{R} used for the calculation of the eddy-induced skew-fluxes are masked at the boundaries 
     1115in exactly the same way as are the triad slopes \rtriad{R} used for the iso-neutral diffusive fluxes, 
     1116as described in \autoref{sec:TRIADS_iso_bdry} and \autoref{fig:TRIADS_bdry_triads}. 
     1117Thus surface layer triads $\triadt{i}{1}{R}{1/2}{-1/2}$ and $\triadt{i+1}{1}{R}{-1/2}{-1/2}$ are masked, 
     1118and both near bottom triad slopes $\triadt{i}{k}{R}{1/2}{1/2}$ and $\triadt{i+1}{k}{R}{-1/2}{1/2}$ are masked when 
     1119either of the $i,k+1$ or $i+1,k+1$ tracer points is masked, \ie\ the $i,k+1$ $u$-point is masked. 
    11201120The namelist parameter \np{ln\_botmix\_triad} has no effect on the eddy-induced skew-fluxes. 
    11211121 
    11221122\subsection{Limiting of the slopes within the interior} 
    1123 \label{sec:limitskew} 
    1124  
    1125 Presently, the iso-neutral slopes $\tilde{r}_i$ relative to geopotentials are limited to be less than $1/100$,  
    1126 exactly as in calculating the iso-neutral diffusion, \S \autoref{sec:limit}.  
     1123\label{sec:TRIADS_limitskew} 
     1124 
     1125Presently, the iso-neutral slopes $\tilde{r}_i$ relative to geopotentials are limited to be less than $1/100$, 
     1126exactly as in calculating the iso-neutral diffusion, \S \autoref{sec:TRIADS_limit}. 
    11271127Each individual triad \rtriadt{R} is so limited. 
    11281128 
    11291129\subsection{Tapering within the surface mixed layer} 
    1130 \label{sec:taperskew} 
    1131  
    1132 The slopes $\tilde{r}_i$ relative to geopotentials (and thus the individual triads \rtriadt{R})  
    1133 are always tapered linearly from their value immediately below the mixed layer to zero at the surface  
    1134 \autoref{eq:rmtilde}, as described in \autoref{sec:lintaper}.  
    1135 This is option (c) of \autoref{fig:eiv_slp}.  
    1136 This linear tapering for the slopes used to calculate the eddy-induced fluxes is unaffected by  
     1130\label{sec:TRIADS_taperskew} 
     1131 
     1132The slopes $\tilde{r}_i$ relative to geopotentials (and thus the individual triads \rtriadt{R}) 
     1133are always tapered linearly from their value immediately below the mixed layer to zero at the surface 
     1134\autoref{eq:TRIADS_rmtilde}, as described in \autoref{sec:TRIADS_lintaper}. 
     1135This is option (c) of \autoref{fig:LDF_eiv_slp}. 
     1136This linear tapering for the slopes used to calculate the eddy-induced fluxes is unaffected by 
    11371137the value of \np{ln\_triad\_iso}. 
    11381138 
     
    11401140the horizontal (the most commonly used options in \NEMO: see \autoref{sec:LDF_coef}), 
    11411141it is equivalent to a horizontal eiv (eddy-induced velocity) that is uniform within the mixed layer 
    1142 \autoref{eq:eiv_v}. 
     1142\autoref{eq:TRIADS_eiv_v}. 
    11431143This ensures that the eiv velocities do not restratify the mixed layer \citep{treguier.held.ea_JPO97,danabasoglu.ferrari.ea_JC08}. 
    11441144Equivantly, in terms of the skew-flux formulation we use here, 
     
    11481148 
    11491149\subsection{Streamfunction diagnostics} 
    1150 \label{sec:sfdiag} 
     1150\label{sec:TRIADS_sfdiag} 
    11511151 
    11521152Where the namelist parameter \np{ln\_traldf\_gdia}\forcode{ = .true.}, 
     
    11541154Each time step, streamfunctions are calculated in the $i$-$k$ and $j$-$k$ planes at 
    11551155$uw$ (integer +1/2 $i$, integer $j$, integer +1/2 $k$) and $vw$ (integer $i$, integer +1/2 $j$, integer +1/2 $k$) 
    1156 points (see Table \autoref{tab:cell}) respectively. 
     1156points (see Table \autoref{tab:DOM_cell}) respectively. 
    11571157We follow \citep{griffies_bk04} and calculate the streamfunction at a given $uw$-point from 
    11581158the surrounding four triads according to: 
    11591159\[ 
    1160   % \label{eq:sfdiagi} 
     1160  % \label{eq:TRIADS_sfdiagi} 
    11611161  {\psi_1}_{i+1/2}^{k+1/2}={\fractext{1}{4}}\sum_{\substack{i_p,\,k_p}} 
    11621162  {A_e}_{i+1/2-i_p}^{k+1/2-k_p}\:\triadd{i+1/2-i_p}{k+1/2-k_p}{R}{i_p}{k_p}. 
    11631163\] 
    11641164The streamfunction $\psi_1$ is calculated similarly at $vw$ points. 
    1165 The eddy-induced velocities are then calculated from the straightforward discretisation of \autoref{eq:eiv_v}: 
    1166 \[ 
    1167   % \label{eq:eiv_v_discrete} 
     1165The eddy-induced velocities are then calculated from the straightforward discretisation of \autoref{eq:TRIADS_eiv_v}: 
     1166\[ 
     1167  % \label{eq:TRIADS_eiv_v_discrete} 
    11681168  \begin{split} 
    11691169    {u^*}_{i+1/2}^{k} & = - \frac{1}{{e_{3u}}_{i}^{k}}\left({\psi_1}_{i+1/2}^{k+1/2}-{\psi_1}_{i+1/2}^{k+1/2}\right),   \\ 
  • NEMO/trunk/doc/latex/NEMO/subfiles/chap_ASM.tex

    r11537 r11543  
    5353additional tendency terms to the prognostic equations: 
    5454\begin{align*} 
    55   % \label{eq:wa_traj_iau} 
     55  % \label{eq:ASM_wa_traj_iau} 
    5656  {\mathbf x}^{a}(t_{i}) = M(t_{i}, t_{0})[{\mathbf x}^{b}(t_{0})] \; + \; F_{i} \delta \tilde{\mathbf x}^{a} 
    5757\end{align*} 
     
    6666The first function (namelist option \np{niaufn}=0) employs constant weights, 
    6767\begin{align} 
    68   \label{eq:F1_i} 
     68  \label{eq:ASM_F1_i} 
    6969  F^{(1)}_{i} 
    7070  =\left\{ 
     
    8080with the weighting reduced linearly to a small value at the window end-points: 
    8181\begin{align} 
    82   \label{eq:F2_i} 
     82  \label{eq:ASM_F2_i} 
    8383  F^{(2)}_{i} 
    8484  =\left\{ 
     
    9292\end{align} 
    9393where $\alpha^{-1} = \sum_{i=1}^{M/2} 2i$ and $M$ is assumed to be even. 
    94 The weights described by \autoref{eq:F2_i} provide a smoother transition of the analysis trajectory from 
    95 one assimilation cycle to the next than that described by \autoref{eq:F1_i}. 
     94The weights described by \autoref{eq:ASM_F2_i} provide a smoother transition of the analysis trajectory from 
     95one assimilation cycle to the next than that described by \autoref{eq:ASM_F1_i}. 
    9696 
    9797%========================================================================== 
     
    106106 
    107107\begin{equation} 
    108   \label{eq:asm_dmp} 
     108  \label{eq:ASM_dmp} 
    109109  \left\{ 
    110110    \begin{aligned} 
     
    120120 
    121121\[ 
    122   % \label{eq:asm_div} 
     122  % \label{eq:ASM_div} 
    123123  \chi^{n-1}_I = \frac{1}{e_{1t}\,e_{2t}\,e_{3t} } 
    124124  \left( {\delta_i \left[ {e_{2u}\,e_{3u}\,u^{n-1}_I} \right] 
     
    126126\] 
    127127 
    128 By the application of \autoref{eq:asm_dmp} the divergence is filtered in each iteration, 
     128By the application of \autoref{eq:ASM_dmp} the divergence is filtered in each iteration, 
    129129and the vorticity is left unchanged. 
    130130In the presence of coastal boundaries with zero velocity increments perpendicular to the coast 
  • NEMO/trunk/doc/latex/NEMO/subfiles/chap_DIA.tex

    r11537 r11543  
    696696\begin{table} 
    697697  \scriptsize 
    698   \begin{tabularx}{\textwidth}{|X|c|c|c|} 
     698  \begin{tabular}{|l|c|c|} 
    699699    \hline 
    700700    tag ids affected by automatic definition of some of their attributes & 
    701701    name attribute                                                       & 
    702     attribute value                      \\ 
     702    attribute value                                                      \\ 
    703703    \hline 
    704704    \hline 
    705705    field\_definition                                                    & 
    706706    freq\_op                                                             & 
    707     \np{rn\_rdt}                         \\ 
     707    \np{rn\_rdt}                                                         \\ 
    708708    \hline 
    709709    SBC                                                                  & 
    710710    freq\_op                                                             & 
    711     \np{rn\_rdt} $\times$ \np{nn\_fsbc}  \\ 
     711    \np{rn\_rdt} $\times$ \np{nn\_fsbc}                                  \\ 
    712712    \hline 
    713713    ptrc\_T                                                              & 
    714714    freq\_op                                                             & 
    715     \np{rn\_rdt} $\times$ \np{nn\_dttrc} \\ 
     715    \np{rn\_rdt} $\times$ \np{nn\_dttrc}                                 \\ 
    716716    \hline 
    717717    diad\_T                                                              & 
    718718    freq\_op                                                             & 
    719     \np{rn\_rdt} $\times$ \np{nn\_dttrc} \\ 
     719    \np{rn\_rdt} $\times$ \np{nn\_dttrc}                                 \\ 
    720720    \hline 
    721721    EqT, EqU, EqW                                                        & 
    722722    jbegin, ni,                                                          & 
    723     according to the grid                \\ 
    724     & 
     723    according to the grid                                                \\ 
     724                                                                         & 
    725725    name\_suffix                                                         & 
    726     \\ 
     726                                                                         \\ 
    727727    \hline 
    728728    TAO, RAMA and PIRATA moorings                                        & 
    729729    zoom\_ibegin, zoom\_jbegin,                                          & 
    730     according to the grid                \\ 
    731     & 
     730    according to the grid                                                \\ 
     731                                                                         & 
    732732    name\_suffix                                                         & 
    733     \\ 
    734     \hline 
    735   \end{tabularx} 
     733                                                                         \\ 
     734    \hline 
     735  \end{tabular} 
    736736\end{table} 
    737737 
     
    739739 
    740740\subsection{XML reference tables} 
    741 \label{subsec:IOM_xmlref} 
     741\label{subsec:DIA_IOM_xmlref} 
    742742 
    743743\begin{enumerate} 
     
    13361336the CF metadata standard. 
    13371337Therefore while a user may wish to add their own metadata to the output files (as demonstrated in example 4 of 
    1338 section \autoref{subsec:IOM_xmlref}) the metadata should, for the most part, comply with the CF-1.5 standard. 
     1338section \autoref{subsec:DIA_IOM_xmlref}) the metadata should, for the most part, comply with the CF-1.5 standard. 
    13391339 
    13401340Some metadata that may significantly increase the file size (horizontal cell areas and vertices) are controlled by 
     
    14071407the mono-processor case (\ie\ global domain of {\small\ttfamily 182x149x31}). 
    14081408An illustration of the potential space savings that NetCDF4 chunking and compression provides is given in 
    1409 table \autoref{tab:NC4} which compares the results of two short runs of the ORCA2\_LIM reference configuration with 
     1409table \autoref{tab:DIA_NC4} which compares the results of two short runs of the ORCA2\_LIM reference configuration with 
    14101410a 4x2 mpi partitioning. 
    14111411Note the variation in the compression ratio achieved which reflects chiefly the dry to wet volume ratio of 
     
    14471447  \end{tabular} 
    14481448  \caption{ 
    1449     \protect\label{tab:NC4} 
     1449    \protect\label{tab:DIA_NC4} 
    14501450    Filesize comparison between NetCDF3 and NetCDF4 with chunking and compression 
    14511451  } 
     
    15151515\section[FLO: On-Line Floats trajectories (\texttt{\textbf{key\_floats}})] 
    15161516{FLO: On-Line Floats trajectories (\protect\key{floats})} 
    1517 \label{sec:FLO} 
     1517\label{sec:DIA_FLO} 
    15181518%--------------------------------------------namflo------------------------------------------------------- 
    15191519 
     
    18471847    \mathcal{V} &=  \mathcal{A}  \;\bar{\eta} 
    18481848  \end{split} 
    1849   \label{eq:MV_nBq} 
     1849  \label{eq:DIA_MV_nBq} 
    18501850\end{equation} 
    18511851 
     
    18551855  \frac{1}{e_3} \partial_t ( e_3\,\rho) + \nabla( \rho \, \textbf{U} ) 
    18561856  = \left. \frac{\textit{emp}}{e_3}\right|_\textit{surface} 
    1857   \label{eq:Co_nBq} 
     1857  \label{eq:DIA_Co_nBq} 
    18581858\end{equation} 
    18591859 
     
    18641864\begin{equation} 
    18651865  \partial_t \mathcal{M} = \mathcal{A} \;\overline{\textit{emp}} 
    1866   \label{eq:Mass_nBq} 
     1866  \label{eq:DIA_Mass_nBq} 
    18671867\end{equation} 
    18681868 
    18691869where $\overline{\textit{emp}} = \int_S \textit{emp}\,ds$ is the net mass flux through the ocean surface. 
    1870 Bringing \autoref{eq:Mass_nBq} and the time derivative of \autoref{eq:MV_nBq} together leads to 
     1870Bringing \autoref{eq:DIA_Mass_nBq} and the time derivative of \autoref{eq:DIA_MV_nBq} together leads to 
    18711871the evolution equation of the mean sea level 
    18721872 
     
    18741874  \partial_t \bar{\eta} = \frac{\overline{\textit{emp}}}{ \bar{\rho}} 
    18751875  - \frac{\mathcal{V}}{\mathcal{A}} \;\frac{\partial_t \bar{\rho} }{\bar{\rho}} 
    1876   \label{eq:ssh_nBq} 
     1876  \label{eq:DIA_ssh_nBq} 
    18771877\end{equation} 
    18781878 
    1879 The first term in equation \autoref{eq:ssh_nBq} alters sea level by adding or subtracting mass from the ocean. 
     1879The first term in equation \autoref{eq:DIA_ssh_nBq} alters sea level by adding or subtracting mass from the ocean. 
    18801880The second term arises from temporal changes in the global mean density; \ie\ from steric effects. 
    18811881 
    18821882In a Boussinesq fluid, $\rho$ is replaced by $\rho_o$ in all the equation except when $\rho$ appears multiplied by 
    18831883the gravity (\ie\ in the hydrostatic balance of the primitive Equations). 
    1884 In particular, the mass conservation equation, \autoref{eq:Co_nBq}, degenerates into the incompressibility equation: 
     1884In particular, the mass conservation equation, \autoref{eq:DIA_Co_nBq}, degenerates into the incompressibility equation: 
    18851885 
    18861886\[ 
    18871887  \frac{1}{e_3} \partial_t ( e_3 ) + \nabla( \textbf{U} ) = \left. \frac{\textit{emp}}{\rho_o \,e_3}\right|_ \textit{surface} 
    1888   % \label{eq:Co_Bq} 
     1888  % \label{eq:DIA_Co_Bq} 
    18891889\] 
    18901890 
     
    18931893\[ 
    18941894  \partial_t \mathcal{V} = \mathcal{A} \;\frac{\overline{\textit{emp}}}{\rho_o} 
    1895   % \label{eq:V_Bq} 
     1895  % \label{eq:DIA_V_Bq} 
    18961896\] 
    18971897 
     
    19121912\begin{equation} 
    19131913  \mathcal{M}_o = \mathcal{M} + \rho_o \,\eta_s \,\mathcal{A} 
    1914   \label{eq:M_Bq} 
     1914  \label{eq:DIA_M_Bq} 
    19151915\end{equation} 
    19161916 
     
    19191919Introducing the total density anomaly, $\mathcal{D}= \int_D d_a \,dv$, 
    19201920where $d_a = (\rho -\rho_o ) / \rho_o$ is the density anomaly used in \NEMO\ (cf. \autoref{subsec:TRA_eos}) 
    1921 in \autoref{eq:M_Bq} leads to a very simple form for the steric height: 
     1921in \autoref{eq:DIA_M_Bq} leads to a very simple form for the steric height: 
    19221922 
    19231923\begin{equation} 
    19241924  \eta_s = - \frac{1}{\mathcal{A}} \mathcal{D} 
    1925   \label{eq:steric_Bq} 
     1925  \label{eq:DIA_steric_Bq} 
    19261926\end{equation} 
    19271927 
     
    19431943(wetting and drying of grid point is not allowed). 
    19441944 
    1945 Third, the discretisation of \autoref{eq:steric_Bq} depends on the type of free surface which is considered. 
     1945Third, the discretisation of \autoref{eq:DIA_steric_Bq} depends on the type of free surface which is considered. 
    19461946In the non linear free surface case, \ie\ \np{ln\_linssh}\forcode{=.true.}, it is given by 
    19471947 
    19481948\[ 
    19491949  \eta_s = - \frac{ \sum_{i,\,j,\,k} d_a\; e_{1t} e_{2t} e_{3t} }{ \sum_{i,\,j,\,k}       e_{1t} e_{2t} e_{3t} } 
    1950   % \label{eq:discrete_steric_Bq_nfs} 
     1950  % \label{eq:DIA_discrete_steric_Bq_nfs} 
    19511951\] 
    19521952 
     
    19581958  \eta_s = - \frac{ \sum_{i,\,j,\,k} d_a\; e_{1t}e_{2t}e_{3t} + \sum_{i,\,j} d_a\; e_{1t}e_{2t} \eta } 
    19591959                  { \sum_{i,\,j,\,k}       e_{1t}e_{2t}e_{3t} + \sum_{i,\,j}       e_{1t}e_{2t} \eta } 
    1960   % \label{eq:discrete_steric_Bq_fs} 
     1960  % \label{eq:DIA_discrete_steric_Bq_fs} 
    19611961\] 
    19621962 
     
    19781978\[ 
    19791979  \eta_s = - \frac{1}{\mathcal{A}} \int_D d_a(T,S_o,p_o) \,dv 
    1980   % \label{eq:thermosteric_Bq} 
     1980  % \label{eq:DIA_thermosteric_Bq} 
    19811981\] 
    19821982 
     
    20142014    \includegraphics[width=\textwidth]{Fig_mask_subasins} 
    20152015    \caption{ 
    2016       \protect\label{fig:mask_subasins} 
     2016      \protect\label{fig:DIA_mask_subasins} 
    20172017      Decomposition of the World Ocean (here ORCA2) into sub-basin used in to 
    20182018      compute the heat and salt transports as well as the meridional stream-function: 
     
    20452045Pacific and Indo-Pacific Oceans (defined north of 30\deg{S}) as well as for the World Ocean. 
    20462046The sub-basin decomposition requires an input file (\ifile{subbasins}) which contains three 2D mask arrays, 
    2047 the Indo-Pacific mask been deduced from the sum of the Indian and Pacific mask (\autoref{fig:mask_subasins}). 
     2047the Indo-Pacific mask been deduced from the sum of the Indian and Pacific mask (\autoref{fig:DIA_mask_subasins}). 
    20482048 
    20492049%------------------------------------------namptr----------------------------------------- 
     
    20932093\[ 
    20942094  C_u = |u|\frac{\rdt}{e_{1u}}, \quad C_v = |v|\frac{\rdt}{e_{2v}}, \quad C_w = |w|\frac{\rdt}{e_{3w}} 
    2095   % \label{eq:CFL} 
     2095  % \label{eq:DIA_CFL} 
    20962096\] 
    20972097 
  • NEMO/trunk/doc/latex/NEMO/subfiles/chap_DOM.tex

    r11537 r11543  
    2929    {\em 
    3030      Compatibility changes Major simplification has moved many of the options to external domain configuration tools. 
    31       (see \autoref{apdx:DOMAINcfg}) 
     31      (see \autoref{apdx:DOMCFG}) 
    3232    }                                                                                            \\ 
    3333    {\em 3.x} & {\em Rachid Benshila, Gurvan Madec \& S\'{e}bastien Masson} & 
     
    3838\newpage 
    3939 
    40 Having defined the continuous equations in \autoref{chap:PE} and chosen a time discretisation \autoref{chap:STP}, 
     40Having defined the continuous equations in \autoref{chap:MB} and chosen a time discretisation \autoref{chap:TD}, 
    4141we need to choose a grid for spatial discretisation and related numerical algorithms. 
    4242In the present chapter, we provide a general description of the staggered grid used in \NEMO, 
     
    6060    \includegraphics[width=\textwidth]{Fig_cell} 
    6161    \caption{ 
    62       \protect\label{fig:cell} 
     62      \protect\label{fig:DOM_cell} 
    6363      Arrangement of variables. 
    6464      $t$ indicates scalar points where temperature, salinity, density, pressure and 
     
    7676The arrangement of variables is the same in all directions. 
    7777It consists of cells centred on scalar points ($t$, $S$, $p$, $\rho$) with vector points $(u, v, w)$ defined in 
    78 the centre of each face of the cells (\autoref{fig:cell}). 
     78the centre of each face of the cells (\autoref{fig:DOM_cell}). 
    7979This is the generalisation to three dimensions of the well-known ``C'' grid in Arakawa's classification 
    8080\citep{mesinger.arakawa_bk76}. 
     
    8484The ocean mesh (\ie\ the position of all the scalar and vector points) is defined by the transformation that 
    8585gives $(\lambda,\varphi,z)$ as a function of $(i,j,k)$. 
    86 The grid-points are located at integer or integer and a half value of $(i,j,k)$ as indicated on \autoref{tab:cell}. 
     86The grid-points are located at integer or integer and a half value of $(i,j,k)$ as indicated on \autoref{tab:DOM_cell}. 
    8787In all the following, subscripts $u$, $v$, $w$, $f$, $uw$, $vw$ or $fw$ indicate the position of 
    8888the grid-point where the scale factors are defined. 
    89 Each scale factor is defined as the local analytical value provided by \autoref{eq:scale_factors}. 
     89Each scale factor is defined as the local analytical value provided by \autoref{eq:MB_scale_factors}. 
    9090As a result, the mesh on which partial derivatives $\pd[]{\lambda}$, $\pd[]{\varphi}$ and 
    9191$\pd[]{z}$ are evaluated is a uniform mesh with a grid size of unity. 
     
    9595centred finite difference approximation, not from their analytical expression. 
    9696This preserves the symmetry of the discrete set of equations and therefore satisfies many of 
    97 the continuous properties (see \autoref{apdx:C}). 
     97the continuous properties (see \autoref{apdx:INVARIANTS}). 
    9898A similar, related remark can be made about the domain size: 
    9999when needed, an area, volume, or the total ocean depth must be evaluated as the product or sum of the relevant scale factors 
     
    123123    \end{tabular} 
    124124    \caption{ 
    125       \protect\label{tab:cell} 
     125      \protect\label{tab:DOM_cell} 
    126126      Location of grid-points as a function of integer or integer and a half value of the column, line or level. 
    127127      This indexing is only used for the writing of the semi -discrete equations. 
     
    145145secondly, analytical transformations encourage good practice by the definition of smoothly varying grids 
    146146(rather than allowing the user to set arbitrary jumps in thickness between adjacent layers) \citep{treguier.dukowicz.ea_JGR96}. 
    147 An example of the effect of such a choice is shown in \autoref{fig:zgr_e3}. 
     147An example of the effect of such a choice is shown in \autoref{fig:DOM_zgr_e3}. 
    148148%>>>>>>>>>>>>>>>>>>>>>>>>>>>> 
    149149\begin{figure}[!t] 
     
    151151    \includegraphics[width=\textwidth]{Fig_zgr_e3} 
    152152    \caption{ 
    153       \protect\label{fig:zgr_e3} 
     153      \protect\label{fig:DOM_zgr_e3} 
    154154      Comparison of (a) traditional definitions of grid-point position and grid-size in the vertical, 
    155155      and (b) analytically derived grid-point position and scale factors. 
     
    174174the midpoint between them are: 
    175175\begin{alignat*}{2} 
    176   % \label{eq:di_mi} 
     176  % \label{eq:DOM_di_mi} 
    177177  \delta_i [q]      &= &       &q (i + 1/2) - q (i - 1/2) \\ 
    178178  \overline q^{\, i} &= &\big\{ &q (i + 1/2) + q (i - 1/2) \big\} / 2 
     
    180180 
    181181Similar operators are defined with respect to $i + 1/2$, $j$, $j + 1/2$, $k$, and $k + 1/2$. 
    182 Following \autoref{eq:PE_grad} and \autoref{eq:PE_lap}, the gradient of a variable $q$ defined at a $t$-point has 
     182Following \autoref{eq:MB_grad} and \autoref{eq:MB_lap}, the gradient of a variable $q$ defined at a $t$-point has 
    183183its three components defined at $u$-, $v$- and $w$-points while its Laplacian is defined at the $t$-point. 
    184184These operators have the following discrete forms in the curvilinear $s$-coordinates system: 
     
    198198\end{multline*} 
    199199 
    200 Following \autoref{eq:PE_curl} and \autoref{eq:PE_div}, a vector $\vect A = (a_1,a_2,a_3)$ defined at 
     200Following \autoref{eq:MB_curl} and \autoref{eq:MB_div}, a vector $\vect A = (a_1,a_2,a_3)$ defined at 
    201201vector points $(u,v,w)$ has its three curl components defined at $vw$-, $uw$, and $f$-points, and 
    202202its divergence defined at $t$-points: 
     
    255255In other words, the adjoint of the differencing and averaging operators are $\delta_i^* = \delta_{i + 1/2}$ and 
    256256$(\overline{\cdots}^{\, i})^* = \overline{\cdots}^{\, i + 1/2}$, respectively. 
    257 These two properties will be used extensively in the \autoref{apdx:C} to 
     257These two properties will be used extensively in the \autoref{apdx:INVARIANTS} to 
    258258demonstrate integral conservative properties of the discrete formulation chosen. 
    259259 
     
    269269    \includegraphics[width=\textwidth]{Fig_index_hor} 
    270270    \caption{ 
    271       \protect\label{fig:index_hor} 
     271      \protect\label{fig:DOM_index_hor} 
    272272      Horizontal integer indexing used in the \fortran code. 
    273273      The dashed area indicates the cell in which variables contained in arrays have the same $i$- and $j$-indices 
     
    290290\label{subsec:DOM_Num_Index_hor} 
    291291 
    292 The indexing in the horizontal plane has been chosen as shown in \autoref{fig:index_hor}. 
     292The indexing in the horizontal plane has been chosen as shown in \autoref{fig:DOM_index_hor}. 
    293293For an increasing $i$ index ($j$ index), 
    294294the $t$-point and the eastward $u$-point (northward $v$-point) have the same index 
    295 (see the dashed area in \autoref{fig:index_hor}). 
     295(see the dashed area in \autoref{fig:DOM_index_hor}). 
    296296A $t$-point and its nearest north-east $f$-point have the same $i$-and $j$-indices. 
    297297 
     
    306306given in \autoref{subsec:DOM_cell}. 
    307307The sea surface corresponds to the $w$-level $k = 1$, which is the same index as the $t$-level just below 
    308 (\autoref{fig:index_vert}). 
     308(\autoref{fig:DOM_index_vert}). 
    309309The last $w$-level ($k = jpk$) either corresponds to or is below the ocean floor while 
    310 the last $t$-level is always outside the ocean domain (\autoref{fig:index_vert}). 
     310the last $t$-level is always outside the ocean domain (\autoref{fig:DOM_index_vert}). 
    311311Note that a $w$-point and the directly underlaying $t$-point have a common $k$ index 
    312312(\ie\ $t$-points and their nearest $w$-point neighbour in negative index direction), 
    313313in contrast to the indexing on the horizontal plane where the $t$-point has the same index as 
    314314the nearest velocity points in the positive direction of the respective horizontal axis index 
    315 (compare the dashed area in \autoref{fig:index_hor} and \autoref{fig:index_vert}). 
     315(compare the dashed area in \autoref{fig:DOM_index_hor} and \autoref{fig:DOM_index_vert}). 
    316316Since the scale factors are chosen to be strictly positive, 
    317317a \textit{minus sign} is included in the \fortran implementations of 
     
    324324    \includegraphics[width=\textwidth]{Fig_index_vert} 
    325325    \caption{ 
    326       \protect\label{fig:index_vert} 
     326      \protect\label{fig:DOM_index_vert} 
    327327      Vertical integer indexing used in the \fortran code. 
    328328      Note that the $k$-axis is oriented downward. 
     
    363363the model domain itself can be altered by runtime selections. 
    364364The code previously used to perform vertical discretisation has been incorporated into an external tool 
    365 (\path{./tools/DOMAINcfg}) which is briefly described in \autoref{apdx:DOMAINcfg}. 
     365(\path{./tools/DOMAINcfg}) which is briefly described in \autoref{apdx:DOMCFG}. 
    366366 
    367367The next subsections summarise the parameter and fields related to the configuration of the whole model domain. 
     
    418418The values of the geographic longitude and latitude arrays at indices $i,j$ correspond to 
    419419the analytical expressions of the longitude $\lambda$ and latitude $\varphi$ as a function of $(i,j)$, 
    420 evaluated at the values as specified in \autoref{tab:cell} for the respective grid-point position. 
     420evaluated at the values as specified in \autoref{tab:DOM_cell} for the respective grid-point position. 
    421421The calculation of the values of the horizontal scale factor arrays in general additionally involves 
    422422partial derivatives of $\lambda$ and $\varphi$ with respect to $i$ and $j$, 
     
    485485    \includegraphics[width=\textwidth]{Fig_z_zps_s_sps} 
    486486    \caption{ 
    487       \protect\label{fig:z_zps_s_sps} 
     487      \protect\label{fig:DOM_z_zps_s_sps} 
    488488      The ocean bottom as seen by the model: 
    489489      (a) $z$-coordinate with full step, 
     
    510510By default a non-linear free surface is used (\np{ln\_linssh} set to \forcode{=.false.} in \nam{dom}): 
    511511the coordinate follow the time-variation of the free surface so that the transformation is time dependent: 
    512 $z(i,j,k,t)$ (\eg\ \autoref{fig:z_zps_s_sps}f). 
     512$z(i,j,k,t)$ (\eg\ \autoref{fig:DOM_z_zps_s_sps}f). 
    513513When a linear free surface is assumed (\np{ln\_linssh} set to \forcode{=.true.} in \nam{dom}), 
    514514the vertical coordinates are fixed in time, but the seawater can move up and down across the $z_0$ surface 
     
    527527\medskip 
    528528The decision on these choices must be made when the \np{cn\_domcfg} file is constructed. 
    529 Three main choices are offered (\autoref{fig:z_zps_s_sps}a-c): 
     529Three main choices are offered (\autoref{fig:DOM_z_zps_s_sps}a-c): 
    530530 
    531531\begin{itemize} 
     
    536536 
    537537Additionally, hybrid combinations of the three main coordinates are available: 
    538 $s-z$ or $s-zps$ coordinate (\autoref{fig:z_zps_s_sps}d and \autoref{fig:z_zps_s_sps}e). 
     538$s-z$ or $s-zps$ coordinate (\autoref{fig:DOM_z_zps_s_sps}d and \autoref{fig:DOM_z_zps_s_sps}e). 
    539539 
    540540A further choice related to vertical coordinate concerns 
     
    678678\section[Initial state (\textit{istate.F90} and \textit{dtatsd.F90})] 
    679679{Initial state (\protect\mdl{istate} and \protect\mdl{dtatsd})} 
    680 \label{sec:DTA_tsd} 
     680\label{sec:DOM_DTA_tsd} 
    681681%-----------------------------------------namtsd------------------------------------------- 
    682682\nlst{namtsd} 
     
    697697  Initial values for T and S are set via a user supplied \rou{usr\_def\_istate} routine contained in \mdl{userdef\_istate}. 
    698698  The default version sets horizontally uniform T and profiles as used in the GYRE configuration 
    699   (see \autoref{sec:CFG_gyre}). 
     699  (see \autoref{sec:CFGS_gyre}). 
    700700\end{description} 
    701701 
  • NEMO/trunk/doc/latex/NEMO/subfiles/chap_DYN.tex

    r11537 r11543  
    6565%           Horizontal divergence and relative vorticity 
    6666%-------------------------------------------------------------------------------------------------------------- 
    67 \subsection[Horizontal divergence and relative vorticity (\textit{divcur.F90})] 
    68 {Horizontal divergence and relative vorticity (\protect\mdl{divcur})} 
     67\subsection[Horizontal divergence and relative vorticity (\textit{divcur.F90})]{Horizontal divergence and relative vorticity (\protect\mdl{divcur})} 
    6968\label{subsec:DYN_divcur} 
    7069 
    7170The vorticity is defined at an $f$-point (\ie\ corner point) as follows: 
    7271\begin{equation} 
    73   \label{eq:divcur_cur} 
     72  \label{eq:DYN_divcur_cur} 
    7473  \zeta =\frac{1}{e_{1f}\,e_{2f} }\left( {\;\delta_{i+1/2} \left[ {e_{2v}\;v} \right] 
    7574      -\delta_{j+1/2} \left[ {e_{1u}\;u} \right]\;} \right) 
     
    7978It is given by: 
    8079\[ 
    81   % \label{eq:divcur_div} 
     80  % \label{eq:DYN_divcur_div} 
    8281  \chi =\frac{1}{e_{1t}\,e_{2t}\,e_{3t} } 
    8382  \left( {\delta_i \left[ {e_{2u}\,e_{3u}\,u} \right] 
     
    102101%           Sea Surface Height evolution 
    103102%-------------------------------------------------------------------------------------------------------------- 
    104 \subsection[Horizontal divergence and relative vorticity (\textit{sshwzv.F90})] 
    105 {Horizontal divergence and relative vorticity (\protect\mdl{sshwzv})} 
     103\subsection[Horizontal divergence and relative vorticity (\textit{sshwzv.F90})]{Horizontal divergence and relative vorticity (\protect\mdl{sshwzv})} 
    106104\label{subsec:DYN_sshwzv} 
    107105 
    108106The sea surface height is given by: 
    109107\begin{equation} 
    110   \label{eq:dynspg_ssh} 
     108  \label{eq:DYN_spg_ssh} 
    111109  \begin{aligned} 
    112110    \frac{\partial \eta }{\partial t} 
     
    123121\textit{emp} can be written as the evaporation minus precipitation, minus the river runoff. 
    124122The sea-surface height is evaluated using exactly the same time stepping scheme as 
    125 the tracer equation \autoref{eq:tra_nxt}: 
     123the tracer equation \autoref{eq:TRA_nxt}: 
    126124a leapfrog scheme in combination with an Asselin time filter, 
    127 \ie\ the velocity appearing in \autoref{eq:dynspg_ssh} is centred in time (\textit{now} velocity). 
     125\ie\ the velocity appearing in \autoref{eq:DYN_spg_ssh} is centred in time (\textit{now} velocity). 
    128126This is of paramount importance. 
    129127Replacing $T$ by the number $1$ in the tracer equation and summing over the water column must lead to 
     
    134132taking into account the change of the thickness of the levels: 
    135133\begin{equation} 
    136   \label{eq:wzv} 
     134  \label{eq:DYN_wzv} 
    137135  \left\{ 
    138136    \begin{aligned} 
     
    148146re-orientated downward. 
    149147\gmcomment{not sure of this...  to be modified with the change in emp setting} 
    150 In the case of a linear free surface, the time derivative in \autoref{eq:wzv} disappears. 
     148In the case of a linear free surface, the time derivative in \autoref{eq:DYN_wzv} disappears. 
    151149The upper boundary condition applies at a fixed level $z=0$. 
    152150The top vertical velocity is thus equal to the divergence of the barotropic transport 
    153 (\ie\ the first term in the right-hand-side of \autoref{eq:dynspg_ssh}). 
     151(\ie\ the first term in the right-hand-side of \autoref{eq:DYN_spg_ssh}). 
    154152 
    155153Note also that whereas the vertical velocity has the same discrete expression in $z$- and $s$-coordinates, 
     
    157155in the second case, $w$ is the velocity normal to the $s$-surfaces. 
    158156Note also that the $k$-axis is re-orientated downwards in the \fortran code compared to 
    159 the indexing used in the semi-discrete equations such as \autoref{eq:wzv} 
     157the indexing used in the semi-discrete equations such as \autoref{eq:DYN_wzv} 
    160158(see \autoref{subsec:DOM_Num_Index_vertical}). 
    161159 
     
    183181%        Vorticity term 
    184182% ------------------------------------------------------------------------------------------------------------- 
    185 \subsection[Vorticity term (\textit{dynvor.F90})] 
    186 {Vorticity term (\protect\mdl{dynvor})} 
     183\subsection[Vorticity term (\textit{dynvor.F90})]{Vorticity term (\protect\mdl{dynvor})} 
    187184\label{subsec:DYN_vor} 
    188185%------------------------------------------nam_dynvor---------------------------------------------------- 
     
    198195horizontal kinetic energy for the planetary vorticity term (MIX scheme); 
    199196or conserving both the potential enstrophy of horizontally non-divergent flow and horizontal kinetic energy 
    200 (EEN scheme) (see \autoref{subsec:C_vorEEN}). 
     197(EEN scheme) (see \autoref{subsec:INVARIANTS_vorEEN}). 
    201198In the case of ENS, ENE or MIX schemes the land sea mask may be slightly modified to ensure the consistency of 
    202199vorticity term with analytical equations (\np{ln\_dynvor\_con}\forcode{=.true.}). 
     
    206203%                 enstrophy conserving scheme 
    207204%------------------------------------------------------------- 
    208 \subsubsection[Enstrophy conserving scheme (\forcode{ln_dynvor_ens=.true.})] 
    209 {Enstrophy conserving scheme (\protect\np{ln\_dynvor\_ens}\forcode{=.true.})} 
     205\subsubsection[Enstrophy conserving scheme (\forcode{ln_dynvor_ens = .true.})]{Enstrophy conserving scheme (\protect\np{ln\_dynvor\_ens}\forcode{ = .true.})} 
    210206\label{subsec:DYN_vor_ens} 
    211207 
     
    216212It is given by: 
    217213\begin{equation} 
    218   \label{eq:dynvor_ens} 
     214  \label{eq:DYN_vor_ens} 
    219215  \left\{ 
    220216    \begin{aligned} 
     
    230226%                 energy conserving scheme 
    231227%------------------------------------------------------------- 
    232 \subsubsection[Energy conserving scheme (\forcode{ln_dynvor_ene=.true.})] 
    233 {Energy conserving scheme (\protect\np{ln\_dynvor\_ene}\forcode{=.true.})} 
     228\subsubsection[Energy conserving scheme (\forcode{ln_dynvor_ene = .true.})]{Energy conserving scheme (\protect\np{ln\_dynvor\_ene}\forcode{ = .true.})} 
    234229\label{subsec:DYN_vor_ene} 
    235230 
     
    237232It is given by: 
    238233\begin{equation} 
    239   \label{eq:dynvor_ene} 
     234  \label{eq:DYN_vor_ene} 
    240235  \left\{ 
    241236    \begin{aligned} 
     
    251246%                 mix energy/enstrophy conserving scheme 
    252247%------------------------------------------------------------- 
    253 \subsubsection[Mixed energy/enstrophy conserving scheme (\forcode{ln_dynvor_mix=.true.})] 
    254 {Mixed energy/enstrophy conserving scheme (\protect\np{ln\_dynvor\_mix}\forcode{=.true.})} 
     248\subsubsection[Mixed energy/enstrophy conserving scheme (\forcode{ln_dynvor_mix = .true.})]{Mixed energy/enstrophy conserving scheme (\protect\np{ln\_dynvor\_mix}\forcode{ = .true.})} 
    255249\label{subsec:DYN_vor_mix} 
    256250 
    257251For the mixed energy/enstrophy conserving scheme (MIX scheme), a mixture of the two previous schemes is used. 
    258 It consists of the ENS scheme (\autoref{eq:dynvor_ens}) for the relative vorticity term, 
    259 and of the ENE scheme (\autoref{eq:dynvor_ene}) applied to the planetary vorticity term. 
    260 \[ 
    261   % \label{eq:dynvor_mix} 
     252It consists of the ENS scheme (\autoref{eq:DYN_vor_ens}) for the relative vorticity term, 
     253and of the ENE scheme (\autoref{eq:DYN_vor_ene}) applied to the planetary vorticity term. 
     254\[ 
     255  % \label{eq:DYN_vor_mix} 
    262256  \left\{ { 
    263257      \begin{aligned} 
     
    277271%                 energy and enstrophy conserving scheme 
    278272%------------------------------------------------------------- 
    279 \subsubsection[Energy and enstrophy conserving scheme (\forcode{ln_dynvor_een=.true.})] 
    280 {Energy and enstrophy conserving scheme (\protect\np{ln\_dynvor\_een}\forcode{=.true.})} 
     273\subsubsection[Energy and enstrophy conserving scheme (\forcode{ln_dynvor_een = .true.})]{Energy and enstrophy conserving scheme (\protect\np{ln\_dynvor\_een}\forcode{ = .true.})} 
    281274\label{subsec:DYN_vor_een} 
    282275 
     
    297290The idea is to get rid of the double averaging by considering triad combinations of vorticity. 
    298291It is noteworthy that this solution is conceptually quite similar to the one proposed by 
    299 \citep{griffies.gnanadesikan.ea_JPO98} for the discretization of the iso-neutral diffusion operator (see \autoref{apdx:C}). 
     292\citep{griffies.gnanadesikan.ea_JPO98} for the discretization of the iso-neutral diffusion operator (see \autoref{apdx:INVARIANTS}). 
    300293 
    301294The \citet{arakawa.hsu_MWR90} vorticity advection scheme for a single layer is modified 
     
    303296First consider the discrete expression of the potential vorticity, $q$, defined at an $f$-point: 
    304297\[ 
    305   % \label{eq:pot_vor} 
     298  % \label{eq:DYN_pot_vor} 
    306299  q  = \frac{\zeta +f} {e_{3f} } 
    307300\] 
    308 where the relative vorticity is defined by (\autoref{eq:divcur_cur}), 
     301where the relative vorticity is defined by (\autoref{eq:DYN_divcur_cur}), 
    309302the Coriolis parameter is given by $f=2 \,\Omega \;\sin \varphi _f $ and the layer thickness at $f$-points is: 
    310303\begin{equation} 
    311   \label{eq:een_e3f} 
     304  \label{eq:DYN_een_e3f} 
    312305  e_{3f} = \overline{\overline {e_{3t} }} ^{\,i+1/2,j+1/2} 
    313306\end{equation} 
     
    326319% >>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> 
    327320 
    328 A key point in \autoref{eq:een_e3f} is how the averaging in the \textbf{i}- and \textbf{j}- directions is made. 
     321A key point in \autoref{eq:DYN_een_e3f} is how the averaging in the \textbf{i}- and \textbf{j}- directions is made. 
    329322It uses the sum of masked t-point vertical scale factor divided either by the sum of the four t-point masks 
    330323(\np{nn\_een\_e3f}\forcode{=1}), or just by $4$ (\np{nn\_een\_e3f}\forcode{=.true.}). 
     
    340333(\autoref{fig:DYN_een_triad}): 
    341334\begin{equation} 
    342   \label{eq:Q_triads} 
     335  \label{eq:DYN_Q_triads} 
    343336  _i^j \mathbb{Q}^{i_p}_{j_p} 
    344337  = \frac{1}{12} \ \left(   q^{i-i_p}_{j+j_p} + q^{i+j_p}_{j+i_p} + q^{i+i_p}_{j-j_p}  \right) 
     
    348341Finally, the vorticity terms are represented as: 
    349342\begin{equation} 
    350   \label{eq:dynvor_een} 
     343  \label{eq:DYN_vor_een} 
    351344  \left\{ { 
    352345      \begin{aligned} 
     
    361354This EEN scheme in fact combines the conservation properties of the ENS and ENE schemes. 
    362355It conserves both total energy and potential enstrophy in the limit of horizontally nondivergent flow 
    363 (\ie\ $\chi$=$0$) (see \autoref{subsec:C_vorEEN}). 
     356(\ie\ $\chi$=$0$) (see \autoref{subsec:INVARIANTS_vorEEN}). 
    364357Applied to a realistic ocean configuration, it has been shown that it leads to a significant reduction of 
    365358the noise in the vertical velocity field \citep{le-sommer.penduff.ea_OM09}. 
     
    371364%           Kinetic Energy Gradient term 
    372365%-------------------------------------------------------------------------------------------------------------- 
    373 \subsection[Kinetic energy gradient term (\textit{dynkeg.F90})] 
    374 {Kinetic energy gradient term (\protect\mdl{dynkeg})} 
     366\subsection[Kinetic energy gradient term (\textit{dynkeg.F90})]{Kinetic energy gradient term (\protect\mdl{dynkeg})} 
    375367\label{subsec:DYN_keg} 
    376368 
    377 As demonstrated in \autoref{apdx:C}, 
     369As demonstrated in \autoref{apdx:INVARIANTS}, 
    378370there is a single discrete formulation of the kinetic energy gradient term that, 
    379371together with the formulation chosen for the vertical advection (see below), 
    380372conserves the total kinetic energy: 
    381373\[ 
    382   % \label{eq:dynkeg} 
     374  % \label{eq:DYN_keg} 
    383375  \left\{ 
    384376    \begin{aligned} 
     
    392384%           Vertical advection term 
    393385%-------------------------------------------------------------------------------------------------------------- 
    394 \subsection[Vertical advection term (\textit{dynzad.F90})] 
    395 {Vertical advection term (\protect\mdl{dynzad})} 
     386\subsection[Vertical advection term (\textit{dynzad.F90})]{Vertical advection term (\protect\mdl{dynzad})} 
    396387\label{subsec:DYN_zad} 
    397388 
     
    400391conserves the total kinetic energy. 
    401392Indeed, the change of KE due to the vertical advection is exactly balanced by 
    402 the change of KE due to the gradient of KE (see \autoref{apdx:C}). 
    403 \[ 
    404   % \label{eq:dynzad} 
     393the change of KE due to the gradient of KE (see \autoref{apdx:INVARIANTS}). 
     394\[ 
     395  % \label{eq:DYN_zad} 
    405396  \left\{ 
    406397    \begin{aligned} 
     
    439430%           Coriolis plus curvature metric terms 
    440431%-------------------------------------------------------------------------------------------------------------- 
    441 \subsection[Coriolis plus curvature metric terms (\textit{dynvor.F90})] 
    442 {Coriolis plus curvature metric terms (\protect\mdl{dynvor})} 
     432\subsection[Coriolis plus curvature metric terms (\textit{dynvor.F90})]{Coriolis plus curvature metric terms (\protect\mdl{dynvor})} 
    443433\label{subsec:DYN_cor_flux} 
    444434 
     
    447437It is given by: 
    448438\begin{multline*} 
    449   % \label{eq:dyncor_metric} 
     439  % \label{eq:DYN_cor_metric} 
    450440  f+\frac{1}{e_1 e_2 }\left( {v\frac{\partial e_2 }{\partial i}  -  u\frac{\partial e_1 }{\partial j}} \right)  \\ 
    451441  \equiv   f + \frac{1}{e_{1f} e_{2f} } \left( { \ \overline v ^{i+1/2}\delta_{i+1/2} \left[ {e_{2u} } \right] 
     
    453443\end{multline*} 
    454444 
    455 Any of the (\autoref{eq:dynvor_ens}), (\autoref{eq:dynvor_ene}) and (\autoref{eq:dynvor_een}) schemes can be used to 
     445Any of the (\autoref{eq:DYN_vor_ens}), (\autoref{eq:DYN_vor_ene}) and (\autoref{eq:DYN_vor_een}) schemes can be used to 
    456446compute the product of the Coriolis parameter and the vorticity. 
    457 However, the energy-conserving scheme (\autoref{eq:dynvor_een}) has exclusively been used to date. 
     447However, the energy-conserving scheme (\autoref{eq:DYN_vor_een}) has exclusively been used to date. 
    458448This term is evaluated using a leapfrog scheme, \ie\ the velocity is centred in time (\textit{now} velocity). 
    459449 
     
    461451%           Flux form Advection term 
    462452%-------------------------------------------------------------------------------------------------------------- 
    463 \subsection[Flux form advection term (\textit{dynadv.F90})] 
    464 {Flux form advection term (\protect\mdl{dynadv})} 
     453\subsection[Flux form advection term (\textit{dynadv.F90})]{Flux form advection term (\protect\mdl{dynadv})} 
    465454\label{subsec:DYN_adv_flux} 
    466455 
    467456The discrete expression of the advection term is given by: 
    468457\[ 
    469   % \label{eq:dynadv} 
     458  % \label{eq:DYN_adv} 
    470459  \left\{ 
    471460    \begin{aligned} 
     
    495484%                 2nd order centred scheme 
    496485%------------------------------------------------------------- 
    497 \subsubsection[CEN2: $2^{nd}$ order centred scheme (\forcode{ln_dynadv_cen2=.true.})] 
    498 {CEN2: $2^{nd}$ order centred scheme (\protect\np{ln\_dynadv\_cen2}\forcode{=.true.})} 
     486\subsubsection[CEN2: $2^{nd}$ order centred scheme (\forcode{ln_dynadv_cen2 = .true.})]{CEN2: $2^{nd}$ order centred scheme (\protect\np{ln\_dynadv\_cen2}\forcode{ = .true.})} 
    499487\label{subsec:DYN_adv_cen2} 
    500488 
    501489In the centered $2^{nd}$ order formulation, the velocity is evaluated as the mean of the two neighbouring points: 
    502490\begin{equation} 
    503   \label{eq:dynadv_cen2} 
     491  \label{eq:DYN_adv_cen2} 
    504492  \left\{ 
    505493    \begin{aligned} 
     
    519507%                 UBS scheme 
    520508%------------------------------------------------------------- 
    521 \subsubsection[UBS: Upstream Biased Scheme (\forcode{ln_dynadv_ubs=.true.})] 
    522 {UBS: Upstream Biased Scheme (\protect\np{ln\_dynadv\_ubs}\forcode{=.true.})} 
     509\subsubsection[UBS: Upstream Biased Scheme (\forcode{ln_dynadv_ubs = .true.})]{UBS: Upstream Biased Scheme (\protect\np{ln\_dynadv\_ubs}\forcode{ = .true.})} 
    523510\label{subsec:DYN_adv_ubs} 
    524511 
     
    527514For example, the evaluation of $u_T^{ubs} $ is done as follows: 
    528515\begin{equation} 
    529   \label{eq:dynadv_ubs} 
     516  \label{eq:DYN_adv_ubs} 
    530517  u_T^{ubs} =\overline u ^i-\;\frac{1}{6} 
    531518  \begin{cases} 
     
    547534The UBS scheme is not used in all directions. 
    548535In the vertical, the centred $2^{nd}$ order evaluation of the advection is preferred, \ie\ $u_{uw}^{ubs}$ and 
    549 $u_{vw}^{ubs}$ in \autoref{eq:dynadv_cen2} are used. 
     536$u_{vw}^{ubs}$ in \autoref{eq:DYN_adv_cen2} are used. 
    550537UBS is diffusive and is associated with vertical mixing of momentum. \gmcomment{ gm  pursue the 
    551538sentence:Since vertical mixing of momentum is a source term of the TKE equation...  } 
    552539 
    553 For stability reasons, the first term in (\autoref{eq:dynadv_ubs}), 
     540For stability reasons, the first term in (\autoref{eq:DYN_adv_ubs}), 
    554541which corresponds to a second order centred scheme, is evaluated using the \textit{now} velocity (centred in time), 
    555542while the second term, which is the diffusion part of the scheme, 
     
    559546Note that the UBS and QUICK (Quadratic Upstream Interpolation for Convective Kinematics) schemes only differ by 
    560547one coefficient. 
    561 Replacing $1/6$ by $1/8$ in (\autoref{eq:dynadv_ubs}) leads to the QUICK advection scheme \citep{webb.de-cuevas.ea_JAOT98}. 
     548Replacing $1/6$ by $1/8$ in (\autoref{eq:DYN_adv_ubs}) leads to the QUICK advection scheme \citep{webb.de-cuevas.ea_JAOT98}. 
    562549This option is not available through a namelist parameter, since the $1/6$ coefficient is hard coded. 
    563550Nevertheless it is quite easy to make the substitution in the \mdl{dynadv\_ubs} module and obtain a QUICK scheme. 
     
    573560%           Hydrostatic pressure gradient term 
    574561% ================================================================ 
    575 \section[Hydrostatic pressure gradient (\textit{dynhpg.F90})] 
    576 {Hydrostatic pressure gradient (\protect\mdl{dynhpg})} 
     562\section[Hydrostatic pressure gradient (\textit{dynhpg.F90})]{Hydrostatic pressure gradient (\protect\mdl{dynhpg})} 
    577563\label{sec:DYN_hpg} 
    578564%------------------------------------------nam_dynhpg--------------------------------------------------- 
     
    596582%           z-coordinate with full step 
    597583%-------------------------------------------------------------------------------------------------------------- 
    598 \subsection[Full step $Z$-coordinate (\forcode{ln_dynhpg_zco=.true.})] 
    599 {Full step $Z$-coordinate (\protect\np{ln\_dynhpg\_zco}\forcode{=.true.})} 
     584\subsection[Full step $Z$-coordinate (\forcode{ln_dynhpg_zco = .true.})]{Full step $Z$-coordinate (\protect\np{ln\_dynhpg\_zco}\forcode{ = .true.})} 
    600585\label{subsec:DYN_hpg_zco} 
    601586 
     
    607592for $k=km$ (surface layer, $jk=1$ in the code) 
    608593\begin{equation} 
    609   \label{eq:dynhpg_zco_surf} 
     594  \label{eq:DYN_hpg_zco_surf} 
    610595  \left\{ 
    611596    \begin{aligned} 
     
    620605for $1<k<km$ (interior layer) 
    621606\begin{equation} 
    622   \label{eq:dynhpg_zco} 
     607  \label{eq:DYN_hpg_zco} 
    623608  \left\{ 
    624609    \begin{aligned} 
     
    633618\end{equation} 
    634619 
    635 Note that the $1/2$ factor in (\autoref{eq:dynhpg_zco_surf}) is adequate because of the definition of $e_{3w}$ as 
     620Note that the $1/2$ factor in (\autoref{eq:DYN_hpg_zco_surf}) is adequate because of the definition of $e_{3w}$ as 
    636621the vertical derivative of the scale factor at the surface level ($z=0$). 
    637622Note also that in case of variable volume level (\texttt{vvl?} defined), 
    638 the surface pressure gradient is included in \autoref{eq:dynhpg_zco_surf} and 
    639 \autoref{eq:dynhpg_zco} through the space and time variations of the vertical scale factor $e_{3w}$. 
     623the surface pressure gradient is included in \autoref{eq:DYN_hpg_zco_surf} and 
     624\autoref{eq:DYN_hpg_zco} through the space and time variations of the vertical scale factor $e_{3w}$. 
    640625 
    641626%-------------------------------------------------------------------------------------------------------------- 
    642627%           z-coordinate with partial step 
    643628%-------------------------------------------------------------------------------------------------------------- 
    644 \subsection[Partial step $Z$-coordinate (\forcode{ln_dynhpg_zps=.true.})] 
    645 {Partial step $Z$-coordinate (\protect\np{ln\_dynhpg\_zps}\forcode{=.true.})} 
     629\subsection[Partial step $Z$-coordinate (\forcode{ln_dynhpg_zps = .true.})]{Partial step $Z$-coordinate (\protect\np{ln\_dynhpg\_zps}\forcode{ = .true.})} 
    646630\label{subsec:DYN_hpg_zps} 
    647631 
     
    674658$\bullet$ Traditional coding (see for example \citet{madec.delecluse.ea_JPO96}: (\np{ln\_dynhpg\_sco}\forcode{=.true.}) 
    675659\begin{equation} 
    676   \label{eq:dynhpg_sco} 
     660  \label{eq:DYN_hpg_sco} 
    677661  \left\{ 
    678662    \begin{aligned} 
     
    686670 
    687671Where the first term is the pressure gradient along coordinates, 
    688 computed as in \autoref{eq:dynhpg_zco_surf} - \autoref{eq:dynhpg_zco}, 
     672computed as in \autoref{eq:DYN_hpg_zco_surf} - \autoref{eq:DYN_hpg_zco}, 
    689673and $z_T$ is the depth of the $T$-point evaluated from the sum of the vertical scale factors at the $w$-point 
    690674($e_{3w}$). 
     
    698682(\np{ln\_dynhpg\_djc}\forcode{=.true.}) (currently disabled; under development) 
    699683 
    700 Note that expression \autoref{eq:dynhpg_sco} is commonly used when the variable volume formulation is activated 
     684Note that expression \autoref{eq:DYN_hpg_sco} is commonly used when the variable volume formulation is activated 
    701685(\texttt{vvl?}) because in that case, even with a flat bottom, 
    702686the coordinate surfaces are not horizontal but follow the free surface \citep{levier.treguier.ea_rpt07}. 
     
    712696\subsection{Ice shelf cavity} 
    713697\label{subsec:DYN_hpg_isf} 
     698 
    714699Beneath an ice shelf, the total pressure gradient is the sum of the pressure gradient due to the ice shelf load and 
    715700the pressure gradient due to the ocean load (\np{ln\_dynhpg\_isf}\forcode{=.true.}).\\ 
     
    722707A detailed description of this method is described in \citet{losch_JGR08}.\\ 
    723708 
    724 The pressure gradient due to ocean load is computed using the expression \autoref{eq:dynhpg_sco} described in 
     709The pressure gradient due to ocean load is computed using the expression \autoref{eq:DYN_hpg_sco} described in 
    725710\autoref{subsec:DYN_hpg_sco}. 
    726711 
     
    728713%           Time-scheme 
    729714%-------------------------------------------------------------------------------------------------------------- 
    730 \subsection[Time-scheme (\forcode{ln_dynhpg_imp={.true.,.false.}})] 
    731 {Time-scheme (\protect\np{ln\_dynhpg\_imp}\forcode{=.true.,.false.})} 
     715\subsection[Time-scheme (\forcode{ln_dynhpg_imp = .{true,false}.})]{Time-scheme (\protect\np{ln\_dynhpg\_imp}\forcode{ = .\{true,false\}}.)} 
    732716\label{subsec:DYN_hpg_imp} 
    733717 
     
    748732 
    749733\begin{equation} 
    750   \label{eq:dynhpg_lf} 
     734  \label{eq:DYN_hpg_lf} 
    751735  \frac{u^{t+\rdt}-u^{t-\rdt}}{2\rdt} = \;\cdots \; 
    752736  -\frac{1}{\rho_o \,e_{1u} }\delta_{i+1/2} \left[ {p_h^t } \right] 
     
    755739$\bullet$ semi-implicit scheme (\np{ln\_dynhpg\_imp}\forcode{=.true.}): 
    756740\begin{equation} 
    757   \label{eq:dynhpg_imp} 
     741  \label{eq:DYN_hpg_imp} 
    758742  \frac{u^{t+\rdt}-u^{t-\rdt}}{2\rdt} = \;\cdots \; 
    759743  -\frac{1}{4\,\rho_o \,e_{1u} } \delta_{i+1/2} \left[ p_h^{t+\rdt} +2\,p_h^t +p_h^{t-\rdt}  \right] 
    760744\end{equation} 
    761745 
    762 The semi-implicit time scheme \autoref{eq:dynhpg_imp} is made possible without 
     746The semi-implicit time scheme \autoref{eq:DYN_hpg_imp} is made possible without 
    763747significant additional computation since the density can be updated to time level $t+\rdt$ before 
    764748computing the horizontal hydrostatic pressure gradient. 
    765749It can be easily shown that the stability limit associated with the hydrostatic pressure gradient doubles using 
    766 \autoref{eq:dynhpg_imp} compared to that using the standard leapfrog scheme \autoref{eq:dynhpg_lf}. 
    767 Note that \autoref{eq:dynhpg_imp} is equivalent to applying a time filter to the pressure gradient to 
     750\autoref{eq:DYN_hpg_imp} compared to that using the standard leapfrog scheme \autoref{eq:DYN_hpg_lf}. 
     751Note that \autoref{eq:DYN_hpg_imp} is equivalent to applying a time filter to the pressure gradient to 
    768752eliminate high frequency IGWs. 
    769 Obviously, when using \autoref{eq:dynhpg_imp}, 
     753Obviously, when using \autoref{eq:DYN_hpg_imp}, 
    770754the doubling of the time-step is achievable only if no other factors control the time-step, 
    771755such as the stability limits associated with advection or diffusion. 
     
    777761The density used to compute the hydrostatic pressure gradient (whatever the formulation) is evaluated as follows: 
    778762\[ 
    779   % \label{eq:rho_flt} 
     763  % \label{eq:DYN_rho_flt} 
    780764  \rho^t = \rho( \widetilde{T},\widetilde {S},z_t) 
    781765  \quad    \text{with}  \quad 
     
    790774% Surface Pressure Gradient 
    791775% ================================================================ 
    792 \section[Surface pressure gradient (\textit{dynspg.F90})] 
    793 {Surface pressure gradient (\protect\mdl{dynspg})} 
     776\section[Surface pressure gradient (\textit{dynspg.F90})]{Surface pressure gradient (\protect\mdl{dynspg})} 
    794777\label{sec:DYN_spg} 
    795778%-----------------------------------------nam_dynspg---------------------------------------------------- 
     
    799782 
    800783Options are defined through the \nam{dyn\_spg} namelist variables. 
    801 The surface pressure gradient term is related to the representation of the free surface (\autoref{sec:PE_hor_pg}). 
     784The surface pressure gradient term is related to the representation of the free surface (\autoref{sec:MB_hor_pg}). 
    802785The main distinction is between the fixed volume case (linear free surface) and 
    803786the variable volume case (nonlinear free surface, \texttt{vvl?} is defined). 
    804 In the linear free surface case (\autoref{subsec:PE_free_surface}) 
     787In the linear free surface case (\autoref{subsec:MB_free_surface}) 
    805788the vertical scale factors $e_{3}$ are fixed in time, 
    806 while they are time-dependent in the nonlinear case (\autoref{subsec:PE_free_surface}). 
     789while they are time-dependent in the nonlinear case (\autoref{subsec:MB_free_surface}). 
    807790With both linear and nonlinear free surface, external gravity waves are allowed in the equations, 
    808791which imposes a very small time step when an explicit time stepping is used. 
    809792Two methods are proposed to allow a longer time step for the three-dimensional equations: 
    810 the filtered free surface, which is a modification of the continuous equations (see \autoref{eq:PE_flt?}), 
     793the filtered free surface, which is a modification of the continuous equations (see \autoref{eq:MB_flt?}), 
    811794and the split-explicit free surface described below. 
    812795The extra term introduced in the filtered method is calculated implicitly, 
     
    815798 
    816799The form of the surface pressure gradient term depends on how the user wants to 
    817 handle the fast external gravity waves that are a solution of the analytical equation (\autoref{sec:PE_hor_pg}). 
     800handle the fast external gravity waves that are a solution of the analytical equation (\autoref{sec:MB_hor_pg}). 
    818801Three formulations are available, all controlled by a CPP key (ln\_dynspg\_xxx): 
    819802an explicit formulation which requires a small time step; 
     
    829812% Explicit free surface formulation 
    830813%-------------------------------------------------------------------------------------------------------------- 
    831 \subsection[Explicit free surface (\texttt{ln\_dynspg\_exp}\forcode{=.true.})] 
    832 {Explicit free surface (\protect\np{ln\_dynspg\_exp}\forcode{=.true.})} 
     814\subsection[Explicit free surface (\texttt{ln\_dynspg\_exp}\forcode{ = .true.})]{Explicit free surface (\protect\np{ln\_dynspg\_exp}\forcode{ = .true.})} 
    833815\label{subsec:DYN_spg_exp} 
    834816 
     
    839821is thus simply given by : 
    840822\begin{equation} 
    841   \label{eq:dynspg_exp} 
     823  \label{eq:DYN_spg_exp} 
    842824  \left\{ 
    843825    \begin{aligned} 
     
    856838% Split-explict free surface formulation 
    857839%-------------------------------------------------------------------------------------------------------------- 
    858 \subsection[Split-explicit free surface (\texttt{ln\_dynspg\_ts}\forcode{=.true.})] 
    859 {Split-explicit free surface (\protect\np{ln\_dynspg\_ts}\forcode{=.true.})} 
     840\subsection[Split-explicit free surface (\texttt{ln\_dynspg\_ts}\forcode{ = .true.})]{Split-explicit free surface (\protect\np{ln\_dynspg\_ts}\forcode{ = .true.})} 
    860841\label{subsec:DYN_spg_ts} 
    861842%------------------------------------------namsplit----------------------------------------------------------- 
     
    868849The general idea is to solve the free surface equation and the associated barotropic velocity equations with 
    869850a smaller time step than $\rdt$, the time step used for the three dimensional prognostic variables 
    870 (\autoref{fig:DYN_dynspg_ts}). 
     851(\autoref{fig:DYN_spg_ts}). 
    871852The size of the small time step, $\rdt_e$ (the external mode or barotropic time step) is provided through 
    872853the \np{nn\_baro} namelist parameter as: $\rdt_e = \rdt / nn\_baro$. 
     
    879860The barotropic mode solves the following equations: 
    880861% \begin{subequations} 
    881 %  \label{eq:BT} 
    882 \begin{equation} 
    883   \label{eq:BT_dyn} 
     862%  \label{eq:DYN_BT} 
     863\begin{equation} 
     864  \label{eq:DYN_BT_dyn} 
    884865  \frac{\partial {\mathrm \overline{{\mathbf U}}_h} }{\partial t}= 
    885866  -f\;{\mathrm {\mathbf k}}\times {\mathrm \overline{{\mathbf U}}_h} 
     
    887868\end{equation} 
    888869\[ 
    889   % \label{eq:BT_ssh} 
     870  % \label{eq:DYN_BT_ssh} 
    890871  \frac{\partial \eta }{\partial t}=-\nabla \cdot \left[ {\left( {H+\eta } \right) \; {\mathrm{\mathbf \overline{U}}}_h \,} \right]+P-E 
    891872\] 
     
    893874where $\mathrm {\overline{\mathbf G}}$ is a forcing term held constant, containing coupling term between modes, 
    894875surface atmospheric forcing as well as slowly varying barotropic terms not explicitly computed to gain efficiency. 
    895 The third term on the right hand side of \autoref{eq:BT_dyn} represents the bottom stress 
    896 (see section \autoref{sec:ZDF_bfr}), explicitly accounted for at each barotropic iteration. 
     876The third term on the right hand side of \autoref{eq:DYN_BT_dyn} represents the bottom stress 
     877(see section \autoref{sec:ZDF_drg}), explicitly accounted for at each barotropic iteration. 
    897878Temporal discretization of the system above follows a three-time step Generalized Forward Backward algorithm 
    898879detailed in \citet{shchepetkin.mcwilliams_OM05}. 
     
    906887    \includegraphics[width=\textwidth]{Fig_DYN_dynspg_ts} 
    907888    \caption{ 
    908       \protect\label{fig:DYN_dynspg_ts} 
     889      \protect\label{fig:DYN_spg_ts} 
    909890      Schematic of the split-explicit time stepping scheme for the external and internal modes. 
    910891      Time increases to the right. In this particular exemple, 
     
    929910In the default case (\np{ln\_bt\_fw}\forcode{=.true.}), 
    930911the external mode is integrated between \textit{now} and \textit{after} baroclinic time-steps 
    931 (\autoref{fig:DYN_dynspg_ts}a). 
     912(\autoref{fig:DYN_spg_ts}a). 
    932913To avoid aliasing of fast barotropic motions into three dimensional equations, 
    933914time filtering is eventually applied on barotropic quantities (\np{ln\_bt\_av}\forcode{=.true.}). 
     
    11011082% Filtered free surface formulation 
    11021083%-------------------------------------------------------------------------------------------------------------- 
    1103 \subsection[Filtered free surface (\texttt{dynspg\_flt?})] 
    1104 {Filtered free surface (\protect\texttt{dynspg\_flt?})} 
     1084\subsection[Filtered free surface (\texttt{dynspg\_flt?})]{Filtered free surface (\protect\texttt{dynspg\_flt?})} 
    11051085\label{subsec:DYN_spg_fltp} 
    11061086 
    11071087The filtered formulation follows the \citet{roullet.madec_JGR00} implementation. 
    1108 The extra term introduced in the equations (see \autoref{subsec:PE_free_surface}) is solved implicitly. 
     1088The extra term introduced in the equations (see \autoref{subsec:MB_free_surface}) is solved implicitly. 
    11091089The elliptic solvers available in the code are documented in \autoref{chap:MISC}. 
    11101090 
     
    11121092\gmcomment{               %%% copy from chap-model basics 
    11131093  \[ 
    1114     % \label{eq:spg_flt} 
     1094    % \label{eq:DYN_spg_flt} 
    11151095    \frac{\partial {\mathrm {\mathbf U}}_h }{\partial t}= {\mathrm {\mathbf M}} 
    11161096    - g \nabla \left( \tilde{\rho} \ \eta \right) 
     
    11201100  $\widetilde{\rho} = \rho / \rho_o$ is the dimensionless density, 
    11211101  and $\mathrm {\mathbf M}$ represents the collected contributions of the Coriolis, hydrostatic pressure gradient, 
    1122   non-linear and viscous terms in \autoref{eq:PE_dyn}. 
     1102  non-linear and viscous terms in \autoref{eq:MB_dyn}. 
    11231103}   %end gmcomment 
    11241104 
     
    11301110% Lateral diffusion term 
    11311111% ================================================================ 
    1132 \section[Lateral diffusion term and operators (\textit{dynldf.F90})] 
    1133 {Lateral diffusion term and operators (\protect\mdl{dynldf})} 
     1112\section[Lateral diffusion term and operators (\textit{dynldf.F90})]{Lateral diffusion term and operators (\protect\mdl{dynldf})} 
    11341113\label{sec:DYN_ldf} 
    11351114%------------------------------------------nam_dynldf---------------------------------------------------- 
     
    11451124\ie\ the velocity appearing in its expression is the \textit{before} velocity in time, 
    11461125except for the pure vertical component that appears when a tensor of rotation is used. 
    1147 This latter term is solved implicitly together with the vertical diffusion term (see \autoref{chap:STP}). 
     1126This latter term is solved implicitly together with the vertical diffusion term (see \autoref{chap:TD}). 
    11481127 
    11491128At the lateral boundaries either free slip, 
     
    11651144 
    11661145% ================================================================ 
    1167 \subsection[Iso-level laplacian (\forcode{ln_dynldf_lap=.true.})] 
    1168 {Iso-level laplacian operator (\protect\np{ln\_dynldf\_lap}\forcode{=.true.})} 
     1146\subsection[Iso-level laplacian (\forcode{ln_dynldf_lap = .true.})]{Iso-level laplacian operator (\protect\np{ln\_dynldf\_lap}\forcode{ = .true.})} 
    11691147\label{subsec:DYN_ldf_lap} 
    11701148 
    11711149For lateral iso-level diffusion, the discrete operator is: 
    11721150\begin{equation} 
    1173   \label{eq:dynldf_lap} 
     1151  \label{eq:DYN_ldf_lap} 
    11741152  \left\{ 
    11751153    \begin{aligned} 
     
    11841162\end{equation} 
    11851163 
    1186 As explained in \autoref{subsec:PE_ldf}, 
     1164As explained in \autoref{subsec:MB_ldf}, 
    11871165this formulation (as the gradient of a divergence and curl of the vorticity) preserves symmetry and 
    11881166ensures a complete separation between the vorticity and divergence parts of the momentum diffusion. 
     
    11911169%           Rotated laplacian operator 
    11921170%-------------------------------------------------------------------------------------------------------------- 
    1193 \subsection[Rotated laplacian (\forcode{ln_dynldf_iso=.true.})] 
    1194 {Rotated laplacian operator (\protect\np{ln\_dynldf\_iso}\forcode{=.true.})} 
     1171\subsection[Rotated laplacian (\forcode{ln_dynldf_iso = .true.})]{Rotated laplacian operator (\protect\np{ln\_dynldf\_iso}\forcode{ = .true.})} 
    11951172\label{subsec:DYN_ldf_iso} 
    11961173 
     
    12061183The resulting discrete representation is: 
    12071184\begin{equation} 
    1208   \label{eq:dyn_ldf_iso} 
     1185  \label{eq:DYN_ldf_iso} 
    12091186  \begin{split} 
    12101187    D_u^{l\textbf{U}} &= \frac{1}{e_{1u} \, e_{2u} \, e_{3u} } \\ 
     
    12501227%           Iso-level bilaplacian operator 
    12511228%-------------------------------------------------------------------------------------------------------------- 
    1252 \subsection[Iso-level bilaplacian (\forcode{ln_dynldf_bilap=.true.})] 
    1253 {Iso-level bilaplacian operator (\protect\np{ln\_dynldf\_bilap}\forcode{=.true.})} 
     1229\subsection[Iso-level bilaplacian (\forcode{ln_dynldf_bilap = .true.})]{Iso-level bilaplacian operator (\protect\np{ln\_dynldf\_bilap}\forcode{ = .true.})} 
    12541230\label{subsec:DYN_ldf_bilap} 
    12551231 
    1256 The lateral fourth order operator formulation on momentum is obtained by applying \autoref{eq:dynldf_lap} twice. 
     1232The lateral fourth order operator formulation on momentum is obtained by applying \autoref{eq:DYN_ldf_lap} twice. 
    12571233It requires an additional assumption on boundary conditions: 
    12581234the first derivative term normal to the coast depends on the free or no-slip lateral boundary conditions chosen, 
     
    12651241%           Vertical diffusion term 
    12661242% ================================================================ 
    1267 \section[Vertical diffusion term (\textit{dynzdf.F90})] 
    1268 {Vertical diffusion term (\protect\mdl{dynzdf})} 
     1243\section[Vertical diffusion term (\textit{dynzdf.F90})]{Vertical diffusion term (\protect\mdl{dynzdf})} 
    12691244\label{sec:DYN_zdf} 
    12701245%----------------------------------------------namzdf------------------------------------------------------ 
     
    12791254(\np{ln\_zdfexp}\forcode{=.true.}) using a time splitting technique (\np{nn\_zdfexp} $>$ 1) or 
    12801255$(b)$ a backward (or implicit) time differencing scheme (\np{ln\_zdfexp}\forcode{=.false.}) 
    1281 (see \autoref{chap:STP}). 
     1256(see \autoref{chap:TD}). 
    12821257Note that namelist variables \np{ln\_zdfexp} and \np{nn\_zdfexp} apply to both tracers and dynamics. 
    12831258 
    12841259The formulation of the vertical subgrid scale physics is the same whatever the vertical coordinate is. 
    1285 The vertical diffusion operators given by \autoref{eq:PE_zdf} take the following semi-discrete space form: 
    1286 \[ 
    1287   % \label{eq:dynzdf} 
     1260The vertical diffusion operators given by \autoref{eq:MB_zdf} take the following semi-discrete space form: 
     1261\[ 
     1262  % \label{eq:DYN_zdf} 
    12881263  \left\{ 
    12891264    \begin{aligned} 
     
    13031278the vertical turbulent momentum fluxes, 
    13041279\begin{equation} 
    1305   \label{eq:dynzdf_sbc} 
     1280  \label{eq:DYN_zdf_sbc} 
    13061281  \left.{\left( {\frac{A^{vm} }{e_3 }\ \frac{\partial \textbf{U}_h}{\partial k}} \right)} \right|_{z=1} 
    13071282  = \frac{1}{\rho_o} \binom{\tau_u}{\tau_v } 
     
    13161291 
    13171292The turbulent flux of momentum at the bottom of the ocean is specified through a bottom friction parameterisation 
    1318 (see \autoref{sec:ZDF_bfr}) 
     1293(see \autoref{sec:ZDF_drg}) 
    13191294 
    13201295% ================================================================ 
     
    13471322\section{Wetting and drying } 
    13481323\label{sec:DYN_wetdry} 
     1324 
    13491325There are two main options for wetting and drying code (wd): 
    13501326(a) an iterative limiter (il) and (b) a directional limiter (dl). 
     
    13951371%   Iterative limiters 
    13961372%----------------------------------------------------------------------------------------- 
    1397 \subsection[Directional limiter (\textit{wet\_dry.F90})] 
    1398 {Directional limiter (\mdl{wet\_dry})} 
     1373\subsection[Directional limiter (\textit{wet\_dry.F90})]{Directional limiter (\mdl{wet\_dry})} 
    13991374\label{subsec:DYN_wd_directional_limiter} 
     1375 
    14001376The principal idea of the directional limiter is that 
    14011377water should not be allowed to flow out of a dry tracer cell (i.e. one whose water depth is less than \np{rn\_wdmin1}). 
     
    14351411%----------------------------------------------------------------------------------------- 
    14361412 
    1437 \subsection[Iterative limiter (\textit{wet\_dry.F90})] 
    1438 {Iterative limiter (\mdl{wet\_dry})} 
     1413\subsection[Iterative limiter (\textit{wet\_dry.F90})]{Iterative limiter (\mdl{wet\_dry})} 
    14391414\label{subsec:DYN_wd_iterative_limiter} 
    14401415 
    1441 \subsubsection[Iterative flux limiter (\textit{wet\_dry.F90})] 
    1442 {Iterative flux limiter (\mdl{wet\_dry})} 
    1443 \label{subsubsec:DYN_wd_il_spg_limiter} 
     1416\subsubsection[Iterative flux limiter (\textit{wet\_dry.F90})]{Iterative flux limiter (\mdl{wet\_dry})} 
     1417\label{subsec:DYN_wd_il_spg_limiter} 
    14441418 
    14451419The iterative limiter modifies the fluxes across the faces of cells that are either already ``dry'' 
     
    14491423 
    14501424The continuity equation for the total water depth in a column 
    1451 \begin{equation} \label{dyn_wd_continuity} 
    1452  \frac{\partial h}{\partial t} + \mathbf{\nabla.}(h\mathbf{u}) = 0 . 
     1425\begin{equation} 
     1426  \label{eq:DYN_wd_continuity} 
     1427  \frac{\partial h}{\partial t} + \mathbf{\nabla.}(h\mathbf{u}) = 0 . 
    14531428\end{equation} 
    14541429can be written in discrete form  as 
    14551430 
    1456 \begin{align} \label{dyn_wd_continuity_2} 
    1457 \frac{e_1 e_2}{\Delta t} ( h_{i,j}(t_{n+1}) - h_{i,j}(t_e) ) 
    1458 &= - ( \mathrm{flxu}_{i+1,j} - \mathrm{flxu}_{i,j}  + \mathrm{flxv}_{i,j+1} - \mathrm{flxv}_{i,j} ) \\ 
    1459 &= \mathrm{zzflx}_{i,j} . 
     1431\begin{align} 
     1432  \label{eq:DYN_wd_continuity_2} 
     1433  \frac{e_1 e_2}{\Delta t} ( h_{i,j}(t_{n+1}) - h_{i,j}(t_e) ) 
     1434  &= - ( \mathrm{flxu}_{i+1,j} - \mathrm{flxu}_{i,j}  + \mathrm{flxv}_{i,j+1} - \mathrm{flxv}_{i,j} ) \\ 
     1435  &= \mathrm{zzflx}_{i,j} . 
    14601436\end{align} 
    14611437 
     
    14701446(zzflxp) and fluxes that are into the cell (zzflxn).  Clearly 
    14711447 
    1472 \begin{equation} \label{dyn_wd_zzflx_p_n_1} 
    1473 \mathrm{zzflx}_{i,j} = \mathrm{zzflxp}_{i,j} + \mathrm{zzflxn}_{i,j} . 
     1448\begin{equation} 
     1449  \label{eq:DYN_wd_zzflx_p_n_1} 
     1450  \mathrm{zzflx}_{i,j} = \mathrm{zzflxp}_{i,j} + \mathrm{zzflxn}_{i,j} . 
    14741451\end{equation} 
    14751452 
     
    14821459$\mathrm{zcoef}_{i,j}^{(m)}$ such that: 
    14831460 
    1484 \begin{equation} \label{dyn_wd_continuity_coef} 
    1485 \begin{split} 
    1486 \mathrm{zzflxp}^{(m)}_{i,j} =& \mathrm{zcoef}_{i,j}^{(m)} \mathrm{zzflxp}^{(0)}_{i,j} \\ 
    1487 \mathrm{zzflxn}^{(m)}_{i,j} =& \mathrm{zcoef}_{i,j}^{(m)} \mathrm{zzflxn}^{(0)}_{i,j} 
    1488 \end{split} 
     1461\begin{equation} 
     1462  \label{eq:DYN_wd_continuity_coef} 
     1463  \begin{split} 
     1464    \mathrm{zzflxp}^{(m)}_{i,j} =& \mathrm{zcoef}_{i,j}^{(m)} \mathrm{zzflxp}^{(0)}_{i,j} \\ 
     1465    \mathrm{zzflxn}^{(m)}_{i,j} =& \mathrm{zcoef}_{i,j}^{(m)} \mathrm{zzflxn}^{(0)}_{i,j} 
     1466  \end{split} 
    14891467\end{equation} 
    14901468 
     
    14941472The iteration is initialised by setting 
    14951473 
    1496 \begin{equation} \label{dyn_wd_zzflx_initial} 
    1497 \mathrm{zzflxp^{(0)}}_{i,j} = \mathrm{zzflxp}_{i,j} , \quad  \mathrm{zzflxn^{(0)}}_{i,j} = \mathrm{zzflxn}_{i,j} . 
     1474\begin{equation} 
     1475  \label{eq:DYN_wd_zzflx_initial} 
     1476  \mathrm{zzflxp^{(0)}}_{i,j} = \mathrm{zzflxp}_{i,j} , \quad  \mathrm{zzflxn^{(0)}}_{i,j} = \mathrm{zzflxn}_{i,j} . 
    14981477\end{equation} 
    14991478 
    15001479The fluxes out of cell $(i,j)$ are updated at the $m+1$th iteration if the depth of the 
    15011480cell on timestep $t_e$, namely $h_{i,j}(t_e)$, is less than the total flux out of the cell 
    1502 times the timestep divided by the cell area. Using (\ref{dyn_wd_continuity_2}) this 
     1481times the timestep divided by the cell area. Using (\autoref{eq:DYN_wd_continuity_2}) this 
    15031482condition is 
    15041483 
    1505 \begin{equation} \label{dyn_wd_continuity_if} 
    1506 h_{i,j}(t_e)  - \mathrm{rn\_wdmin1} <  \frac{\Delta t}{e_1 e_2} ( \mathrm{zzflxp}^{(m)}_{i,j} + \mathrm{zzflxn}^{(m)}_{i,j} ) . 
    1507 \end{equation} 
    1508  
    1509 Rearranging (\ref{dyn_wd_continuity_if}) we can obtain an expression for the maximum 
     1484\begin{equation} 
     1485  \label{eq:DYN_wd_continuity_if} 
     1486  h_{i,j}(t_e)  - \mathrm{rn\_wdmin1} <  \frac{\Delta t}{e_1 e_2} ( \mathrm{zzflxp}^{(m)}_{i,j} + \mathrm{zzflxn}^{(m)}_{i,j} ) . 
     1487\end{equation} 
     1488 
     1489Rearranging (\autoref{eq:DYN_wd_continuity_if}) we can obtain an expression for the maximum 
    15101490outward flux that can be allowed and still maintain the minimum wet depth: 
    15111491 
    1512 \begin{equation} \label{dyn_wd_max_flux} 
    1513 \begin{split} 
    1514 \mathrm{zzflxp}^{(m+1)}_{i,j} = \Big[ (h_{i,j}(t_e) & - \mathrm{rn\_wdmin1} - \mathrm{rn\_wdmin2})  \frac{e_1 e_2}{\Delta t} \phantom{]} \\ 
    1515 \phantom{[} & -  \mathrm{zzflxn}^{(m)}_{i,j} \Big] 
    1516 \end{split} 
     1492\begin{equation} 
     1493  \label{eq:DYN_wd_max_flux} 
     1494  \begin{split} 
     1495    \mathrm{zzflxp}^{(m+1)}_{i,j} = \Big[ (h_{i,j}(t_e) & - \mathrm{rn\_wdmin1} - \mathrm{rn\_wdmin2})  \frac{e_1 e_2}{\Delta t} \phantom{]} \\ 
     1496    \phantom{[} & -  \mathrm{zzflxn}^{(m)}_{i,j} \Big] 
     1497  \end{split} 
    15171498\end{equation} 
    15181499 
    15191500Note a small tolerance ($\mathrm{rn\_wdmin2}$) has been introduced here {\itshape [Q: Why is 
    1520 this necessary/desirable?]}. Substituting from (\ref{dyn_wd_continuity_coef}) gives an 
     1501this necessary/desirable?]}. Substituting from (\autoref{eq:DYN_wd_continuity_coef}) gives an 
    15211502expression for the coefficient needed to multiply the outward flux at this cell in order 
    15221503to avoid drying. 
    15231504 
    1524 \begin{equation} \label{dyn_wd_continuity_nxtcoef} 
    1525 \begin{split} 
    1526 \mathrm{zcoef}^{(m+1)}_{i,j} = \Big[ (h_{i,j}(t_e) & - \mathrm{rn\_wdmin1} - \mathrm{rn\_wdmin2})  \frac{e_1 e_2}{\Delta t} \phantom{]} \\ 
    1527 \phantom{[} & -  \mathrm{zzflxn}^{(m)}_{i,j} \Big] \frac{1}{ \mathrm{zzflxp}^{(0)}_{i,j} } 
    1528 \end{split} 
     1505\begin{equation} 
     1506  \label{eq:DYN_wd_continuity_nxtcoef} 
     1507  \begin{split} 
     1508    \mathrm{zcoef}^{(m+1)}_{i,j} = \Big[ (h_{i,j}(t_e) & - \mathrm{rn\_wdmin1} - \mathrm{rn\_wdmin2})  \frac{e_1 e_2}{\Delta t} \phantom{]} \\ 
     1509    \phantom{[} & -  \mathrm{zzflxn}^{(m)}_{i,j} \Big] \frac{1}{ \mathrm{zzflxp}^{(0)}_{i,j} } 
     1510  \end{split} 
    15291511\end{equation} 
    15301512 
     
    15451527%      Surface pressure gradients 
    15461528%---------------------------------------------------------------------------------------- 
    1547 \subsubsection[Modification of surface pressure gradients (\textit{dynhpg.F90})] 
    1548 {Modification of surface pressure gradients (\mdl{dynhpg})} 
    1549 \label{subsubsec:DYN_wd_il_spg} 
     1529\subsubsection[Modification of surface pressure gradients (\textit{dynhpg.F90})]{Modification of surface pressure gradients (\mdl{dynhpg})} 
     1530\label{subsec:DYN_wd_il_spg} 
    15501531 
    15511532At ``dry'' points the water depth is usually close to $\mathrm{rn\_wdmin1}$. If the 
     
    15601541neighbouring $(i+1,j)$ and $(i,j)$ tracer points.  zcpx is calculated using two logicals 
    15611542variables, $\mathrm{ll\_tmp1}$ and $\mathrm{ll\_tmp2}$ which are evaluated for each grid 
    1562 column.  The three possible combinations are illustrated in figure \ref{Fig_WAD_dynhpg}. 
     1543column.  The three possible combinations are illustrated in figure \autoref{fig:DYN_WAD_dynhpg}. 
    15631544 
    15641545%>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> 
    15651546\begin{figure}[!ht] \begin{center} 
    15661547\includegraphics[width=\textwidth]{Fig_WAD_dynhpg} 
    1567 \caption{ \label{Fig_WAD_dynhpg} 
    1568 Illustrations of the three possible combinations of the logical variables controlling the 
    1569 limiting of the horizontal pressure gradient in wetting and drying regimes} 
    1570 \end{center}\end{figure} 
     1548\caption{ 
     1549  \label{fig:DYN_WAD_dynhpg} 
     1550  Illustrations of the three possible combinations of the logical variables controlling the 
     1551  limiting of the horizontal pressure gradient in wetting and drying regimes} 
     1552\end{center} 
     1553\end{figure} 
    15711554%>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> 
    15721555 
     
    15761559of the topography at the two points: 
    15771560 
    1578 \begin{equation} \label{dyn_ll_tmp1} 
    1579 \begin{split} 
    1580 \mathrm{ll\_tmp1}  = & \mathrm{MIN(sshn(ji,jj), sshn(ji+1,jj))} > \\ 
     1561\begin{equation} 
     1562  \label{eq:DYN_ll_tmp1} 
     1563  \begin{split} 
     1564    \mathrm{ll\_tmp1}  = & \mathrm{MIN(sshn(ji,jj), sshn(ji+1,jj))} > \\ 
    15811565                     & \quad \mathrm{MAX(-ht\_wd(ji,jj), -ht\_wd(ji+1,jj))\  .and.} \\ 
    1582 & \mathrm{MAX(sshn(ji,jj) + ht\_wd(ji,jj),} \\ 
    1583 & \mathrm{\phantom{MAX(}sshn(ji+1,jj) + ht\_wd(ji+1,jj))} >\\ 
    1584 & \quad\quad\mathrm{rn\_wdmin1 + rn\_wdmin2 } 
    1585 \end{split} 
     1566                     & \mathrm{MAX(sshn(ji,jj) + ht\_wd(ji,jj),} \\ 
     1567                     & \mathrm{\phantom{MAX(}sshn(ji+1,jj) + ht\_wd(ji+1,jj))} >\\ 
     1568                     & \quad\quad\mathrm{rn\_wdmin1 + rn\_wdmin2 } 
     1569  \end{split} 
    15861570\end{equation} 
    15871571 
     
    15901574at the two points plus $\mathrm{rn\_wdmin1} + \mathrm{rn\_wdmin2}$ 
    15911575 
    1592 \begin{equation} \label{dyn_ll_tmp2} 
    1593 \begin{split} 
    1594 \mathrm{ ll\_tmp2 } = & \mathrm{( ABS( sshn(ji,jj) - sshn(ji+1,jj) ) > 1.E-12 )\ .AND.}\\ 
    1595 & \mathrm{( MAX(sshn(ji,jj), sshn(ji+1,jj)) > } \\ 
    1596 & \mathrm{\phantom{(} MAX(-ht\_wd(ji,jj), -ht\_wd(ji+1,jj)) + rn\_wdmin1 + rn\_wdmin2}) . 
    1597 \end{split} 
     1576\begin{equation} 
     1577  \label{eq:DYN_ll_tmp2} 
     1578  \begin{split} 
     1579    \mathrm{ ll\_tmp2 } = & \mathrm{( ABS( sshn(ji,jj) - sshn(ji+1,jj) ) > 1.E-12 )\ .AND.}\\ 
     1580    & \mathrm{( MAX(sshn(ji,jj), sshn(ji+1,jj)) > } \\ 
     1581    & \mathrm{\phantom{(} MAX(-ht\_wd(ji,jj), -ht\_wd(ji+1,jj)) + rn\_wdmin1 + rn\_wdmin2}) . 
     1582  \end{split} 
    15981583\end{equation} 
    15991584 
     
    16111596conditions. 
    16121597 
    1613 \subsubsection[Additional considerations (\textit{usrdef\_zgr.F90})] 
    1614 {Additional considerations (\mdl{usrdef\_zgr})} 
    1615 \label{subsubsec:WAD_additional} 
     1598\subsubsection[Additional considerations (\textit{usrdef\_zgr.F90})]{Additional considerations (\mdl{usrdef\_zgr})} 
     1599\label{subsec:DYN_WAD_additional} 
    16161600 
    16171601In the very shallow water where wetting and drying occurs the parametrisation of 
     
    16261610%      The WAD test cases 
    16271611%---------------------------------------------------------------------------------------- 
    1628 \subsection[The WAD test cases (\textit{usrdef\_zgr.F90})] 
    1629 {The WAD test cases (\mdl{usrdef\_zgr})} 
    1630 \label{WAD_test_cases} 
     1612\subsection[The WAD test cases (\textit{usrdef\_zgr.F90})]{The WAD test cases (\mdl{usrdef\_zgr})} 
     1613\label{subsec:DYN_WAD_test_cases} 
    16311614 
    16321615See the WAD tests MY\_DOC documention for details of the WAD test cases. 
     
    16371620% Time evolution term 
    16381621% ================================================================ 
    1639 \section[Time evolution term (\textit{dynnxt.F90})] 
    1640 {Time evolution term (\protect\mdl{dynnxt})} 
     1622\section[Time evolution term (\textit{dynnxt.F90})]{Time evolution term (\protect\mdl{dynnxt})} 
    16411623\label{sec:DYN_nxt} 
    16421624 
     
    16481630Options are defined through the \nam{dom} namelist variables. 
    16491631The general framework for dynamics time stepping is a leap-frog scheme, 
    1650 \ie\ a three level centred time scheme associated with an Asselin time filter (cf. \autoref{chap:STP}). 
     1632\ie\ a three level centred time scheme associated with an Asselin time filter (cf. \autoref{chap:TD}). 
    16511633The scheme is applied to the velocity, except when 
    16521634using the flux form of momentum advection (cf. \autoref{sec:DYN_adv_cor_flux}) 
    16531635in the variable volume case (\texttt{vvl?} defined), 
    1654 where it has to be applied to the thickness weighted velocity (see \autoref{sec:A_momentum}) 
     1636where it has to be applied to the thickness weighted velocity (see \autoref{sec:SCOORD_momentum}) 
    16551637 
    16561638$\bullet$ vector invariant form or linear free surface 
    16571639(\np{ln\_dynhpg\_vec}\forcode{=.true.} ; \texttt{vvl?} not defined): 
    16581640\[ 
    1659   % \label{eq:dynnxt_vec} 
     1641  % \label{eq:DYN_nxt_vec} 
    16601642  \left\{ 
    16611643    \begin{aligned} 
     
    16691651(\np{ln\_dynhpg\_vec}\forcode{=.false.} ; \texttt{vvl?} defined): 
    16701652\[ 
    1671   % \label{eq:dynnxt_flux} 
     1653  % \label{eq:DYN_nxt_flux} 
    16721654  \left\{ 
    16731655    \begin{aligned} 
  • NEMO/trunk/doc/latex/NEMO/subfiles/chap_LBC.tex

    r11537 r11543  
    5757 
    5858\[ 
    59   % \label{eq:lbc_aaaa} 
     59  % \label{eq:LBC_aaaa} 
    6060  \frac{A^{lT} }{e_1 }\frac{\partial T}{\partial i}\equiv \frac{A_u^{lT} 
    6161  }{e_{1u} } \; \delta_{i+1 / 2} \left[ T \right]\;\;mask_u 
     
    134134  the no-slip boundary condition, simply by multiplying it by the mask$_{f}$ : 
    135135  \[ 
    136     % \label{eq:lbc_bbbb} 
     136    % \label{eq:LBC_bbbb} 
    137137    \zeta \equiv \frac{1}{e_{1f} {\kern 1pt}e_{2f} }\left( {\delta_{i+1/2} 
    138138        \left[ {e_{2v} \,v} \right]-\delta_{j+1/2} \left[ {e_{1u} \,u} \right]} 
     
    226226The north fold boundary condition has been introduced in order to handle the north boundary of 
    227227a three-polar ORCA grid. 
    228 Such a grid has two poles in the northern hemisphere (\autoref{fig:MISC_ORCA_msh}, 
    229 and thus requires a specific treatment illustrated in \autoref{fig:North_Fold_T}. 
     228Such a grid has two poles in the northern hemisphere (\autoref{fig:CFGS_ORCA_msh}, 
     229and thus requires a specific treatment illustrated in \autoref{fig:LBC_North_Fold_T}. 
    230230Further information can be found in \mdl{lbcnfd} module which applies the north fold boundary condition. 
    231231 
     
    235235    \includegraphics[width=\textwidth]{Fig_North_Fold_T} 
    236236    \caption{ 
    237       \protect\label{fig:North_Fold_T} 
     237      \protect\label{fig:LBC_North_Fold_T} 
    238238      North fold boundary with a $T$-point pivot and cyclic east-west boundary condition ($jperio=4$), 
    239239      as used in ORCA 2, 1/4, and 1/12. 
     
    256256%----------------------------------------------------------------------------------------------- 
    257257 
    258 For massively parallel processing (mpp), a domain decomposition method is used. The basic idea of the method is to split the large computation domain of a numerical experiment into several smaller domains and solve the set of equations by addressing independent local problems. Each processor has its own local memory and computes the model equation over a subdomain of the whole model domain. The subdomain boundary conditions are specified through communications between processors which are organized by explicit statements (message passing method). The present implementation is largely inspired by Guyon's work [Guyon 1995]. 
     258For massively parallel processing (mpp), a domain decomposition method is used. 
     259The basic idea of the method is to split the large computation domain of a numerical experiment into several smaller domains and 
     260solve the set of equations by addressing independent local problems. 
     261Each processor has its own local memory and computes the model equation over a subdomain of the whole model domain. 
     262The subdomain boundary conditions are specified through communications between processors which are organized by 
     263explicit statements (message passing method). 
     264The present implementation is largely inspired by Guyon's work [Guyon 1995]. 
    259265 
    260266The parallelization strategy is defined by the physical characteristics of the ocean model. 
     
    272278each processor sends to its neighbouring processors the update values of the points corresponding to 
    273279the interior overlapping area to its neighbouring sub-domain (\ie\ the innermost of the two overlapping rows). 
    274 Communications are first done according to the east-west direction and next according to the north-south direction. There is no specific communications for the corners. The communication is done through the Message Passing Interface (MPI) and requires \key{mpp\_mpi}. Use also \key{mpi2} if MPI3 is not available on your computer. 
     280Communications are first done according to the east-west direction and next according to the north-south direction. 
     281There is no specific communications for the corners. 
     282The communication is done through the Message Passing Interface (MPI) and requires \key{mpp\_mpi}. 
     283Use also \key{mpi2} if MPI3 is not available on your computer. 
    275284The data exchanges between processors are required at the very place where 
    276285lateral domain boundary conditions are set in the mono-domain computation: 
     
    285294    \includegraphics[width=\textwidth]{Fig_mpp} 
    286295    \caption{ 
    287       \protect\label{fig:mpp} 
     296      \protect\label{fig:LBC_mpp} 
    288297      Positioning of a sub-domain when massively parallel processing is used. 
    289298    } 
     
    292301%>>>>>>>>>>>>>>>>>>>>>>>>>>>> 
    293302 
    294 In \NEMO, the splitting is regular and arithmetic. The total number of subdomains corresponds to the number of MPI processes allocated to \NEMO\ when the model is launched (\ie\ mpirun -np x ./nemo will automatically give x subdomains). The i-axis is divided by \np{jpni} and the j-axis by \np{jpnj}. These parameters are defined in \nam{mpp} namelist. If \np{jpni} and \np{jpnj} are < 1, they will be automatically redefined in the code to give the best domain decomposition (see bellow). 
    295  
    296 Each processor is independent and without message passing or synchronous process, programs run alone and access just its own local memory. For this reason, the main model dimensions are now the local dimensions of the subdomain (pencil) that are named \jp{jpi}, \jp{jpj}, \jp{jpk}. 
    297 These dimensions include the internal domain and the overlapping rows. The number of rows to exchange (known as the halo) is usually set to one (nn\_hls=1, in \mdl{par\_oce}, and must be kept to one until further notice). The whole domain dimensions are named \jp{jpiglo}, \jp{jpjglo} and \jp{jpk}. The relationship between the whole domain and a sub-domain is: 
    298 \[ 
    299   jpi = ( jpiglo-2\times nn\_hls + (jpni-1) ) / jpni + 2\times nn\_hls 
    300 \] 
    301 \[ 
     303In \NEMO, the splitting is regular and arithmetic. 
     304The total number of subdomains corresponds to the number of MPI processes allocated to \NEMO\ when the model is launched 
     305(\ie\ mpirun -np x ./nemo will automatically give x subdomains). 
     306The i-axis is divided by \np{jpni} and the j-axis by \np{jpnj}. 
     307These parameters are defined in \nam{mpp} namelist. 
     308If \np{jpni} and \np{jpnj} are < 1, they will be automatically redefined in the code to give the best domain decomposition 
     309(see bellow). 
     310 
     311Each processor is independent and without message passing or synchronous process, programs run alone and access just its own local memory. 
     312For this reason, 
     313the main model dimensions are now the local dimensions of the subdomain (pencil) that are named \jp{jpi}, \jp{jpj}, \jp{jpk}. 
     314These dimensions include the internal domain and the overlapping rows. 
     315The number of rows to exchange (known as the halo) is usually set to one (nn\_hls=1, in \mdl{par\_oce}, 
     316and must be kept to one until further notice). 
     317The whole domain dimensions are named \jp{jpiglo}, \jp{jpjglo} and \jp{jpk}. 
     318The relationship between the whole domain and a sub-domain is: 
     319\begin{gather*} 
     320  jpi = ( jpiglo-2\times nn\_hls + (jpni-1) ) / jpni + 2\times nn\_hls \\ 
    302321  jpj = ( jpjglo-2\times nn\_hls + (jpnj-1) ) / jpnj + 2\times nn\_hls 
    303 \] 
    304  
    305 One also defines variables nldi and nlei which correspond to the internal domain bounds, and the variables nimpp and njmpp which are the position of the (1,1) grid-point in the global domain (\autoref{fig:mpp}). Note that since the version 4, there is no more extra-halo area as defined in \autoref{fig:mpp} so \jp{jpi} is now always equal to nlci and \jp{jpj} equal to nlcj. 
     322\end{gather*} 
     323 
     324One also defines variables nldi and nlei which correspond to the internal domain bounds, and the variables nimpp and njmpp which are the position of the (1,1) grid-point in the global domain (\autoref{fig:LBC_mpp}). Note that since the version 4, there is no more extra-halo area as defined in \autoref{fig:LBC_mpp} so \jp{jpi} is now always equal to nlci and \jp{jpj} equal to nlcj. 
    306325 
    307326An element of $T_{l}$, a local array (subdomain) corresponds to an element of $T_{g}$, 
    308327a global array (whole domain) by the relationship: 
    309328\[ 
    310   % \label{eq:lbc_nimpp} 
     329  % \label{eq:LBC_nimpp} 
    311330  T_{g} (i+nimpp-1,j+njmpp-1,k) = T_{l} (i,j,k), 
    312331\] 
     
    322341 
    323342If the domain decomposition is automatically defined (when \np{jpni} and \np{jpnj} are < 1), the decomposition chosen by the model will minimise the sub-domain size (defined as $max_{all domains}(jpi \times jpj)$) and maximize the number of eliminated land subdomains. This means that no other domain decomposition (a set of \np{jpni} and \np{jpnj} values) will use less processes than $(jpni  \times  jpnj - N_{land})$ and get a smaller subdomain size. 
    324 In order to specify $N_{mpi}$ properly (minimize $N_{useless}$), you must run the model once with \np{ln\_list} activated. In this case, the model will start the initialisation phase, print the list of optimum decompositions ($N_{mpi}$, \np{jpni} and \np{jpnj}) in \texttt{ocean.output} and directly abort. The maximum value of $N_{mpi}$ tested in this list is given by $max(N_{MPI\_tasks}, \np{jpni} \times \np{jpnj})$. For example, run the model on 40 nodes with ln\_list activated and $\np{jpni} = 10000$ and $\np{jpnj} = 1$, will print the list of optimum domains decomposition from 1 to about 10000.  
    325  
    326 Processors are numbered from 0 to $N_{mpi} - 1$. Subdomains containning some ocean points are numbered first from 0 to $jpni * jpnj - N_{land} -1$. The remaining $N_{useless}$ land subdomains are numbered next, which means that, for a given (\np{jpni}, \np{jpnj}), the numbers attributed to he ocean subdomains do not vary with $N_{useless}$.  
     343In order to specify $N_{mpi}$ properly (minimize $N_{useless}$), you must run the model once with \np{ln\_list} activated. In this case, the model will start the initialisation phase, print the list of optimum decompositions ($N_{mpi}$, \np{jpni} and \np{jpnj}) in \texttt{ocean.output} and directly abort. The maximum value of $N_{mpi}$ tested in this list is given by $max(N_{MPI\_tasks}, \np{jpni} \times \np{jpnj})$. For example, run the model on 40 nodes with ln\_list activated and $\np{jpni} = 10000$ and $\np{jpnj} = 1$, will print the list of optimum domains decomposition from 1 to about 10000. 
     344 
     345Processors are numbered from 0 to $N_{mpi} - 1$. Subdomains containning some ocean points are numbered first from 0 to $jpni * jpnj - N_{land} -1$. The remaining $N_{useless}$ land subdomains are numbered next, which means that, for a given (\np{jpni}, \np{jpnj}), the numbers attributed to he ocean subdomains do not vary with $N_{useless}$. 
    327346 
    328347When land processors are eliminated, the value corresponding to these locations in the model output files is undefined. \np{ln\_mskland} must be activated in order avoid Not a Number values in output files. Note that it is better to not eliminate land processors when creating a meshmask file (\ie\ when setting a non-zero value to \np{nn\_msh}). 
     
    332351  \begin{center} 
    333352    \includegraphics[width=\textwidth]{Fig_mppini2} 
    334     \caption { 
    335       \protect\label{fig:mppini2} 
     353    \caption[Atlantic domain]{ 
     354      \protect\label{fig:LBC_mppini2} 
    336355      Example of Atlantic domain defined for the CLIPPER projet. 
    337356      Initial grid is composed of 773 x 1236 horizontal points. 
     
    374393%---------------------------------------------- 
    375394\subsection{Namelists} 
    376 \label{subsec:BDY_namelist} 
     395\label{subsec:LBC_bdy_namelist} 
    377396 
    378397The BDY module is activated by setting \np{ln\_bdy}\forcode{=.true.} . 
     
    384403In the example above, there are two boundary sets, the first of which is defined via a file and 
    385404the second is defined in the namelist. 
    386 For more details of the definition of the boundary geometry see section \autoref{subsec:BDY_geometry}. 
     405For more details of the definition of the boundary geometry see section \autoref{subsec:LBC_bdy_geometry}. 
    387406 
    388407For each boundary set a boundary condition has to be chosen for the barotropic solution 
     
    441460%---------------------------------------------- 
    442461\subsection{Flow relaxation scheme} 
    443 \label{subsec:BDY_FRS_scheme} 
     462\label{subsec:LBC_bdy_FRS_scheme} 
    444463 
    445464The Flow Relaxation Scheme (FRS) \citep{davies_QJRMS76,engedahl_T95}, 
     
    448467Given a model prognostic variable $\Phi$ 
    449468\[ 
    450   % \label{eq:bdy_frs1} 
     469  % \label{eq:LBC_bdy_frs1} 
    451470  \Phi(d) = \alpha(d)\Phi_{e}(d) + (1-\alpha(d))\Phi_{m}(d)\;\;\;\;\; d=1,N 
    452471\] 
     
    457476the prognostic equation for $\Phi$ of the form: 
    458477\[ 
    459   % \label{eq:bdy_frs2} 
     478  % \label{eq:LBC_bdy_frs2} 
    460479  -\frac{1}{\tau}\left(\Phi - \Phi_{e}\right) 
    461480\] 
    462481where the relaxation time scale $\tau$ is given by a function of $\alpha$ and the model time step $\Delta t$: 
    463482\[ 
    464   % \label{eq:bdy_frs3} 
     483  % \label{eq:LBC_bdy_frs3} 
    465484  \tau = \frac{1-\alpha}{\alpha}  \,\rdt 
    466485\] 
     
    472491The function $\alpha$ is specified as a $tanh$ function: 
    473492\[ 
    474   % \label{eq:bdy_frs4} 
     493  % \label{eq:LBC_bdy_frs4} 
    475494  \alpha(d) = 1 - \tanh\left(\frac{d-1}{2}\right),       \quad d=1,N 
    476495\] 
     
    480499%---------------------------------------------- 
    481500\subsection{Flather radiation scheme} 
    482 \label{subsec:BDY_flather_scheme} 
     501\label{subsec:LBC_bdy_flather_scheme} 
    483502 
    484503The \citet{flather_JPO94} scheme is a radiation condition on the normal, 
    485504depth-mean transport across the open boundary. 
    486505It takes the form 
    487 \begin{equation}  \label{eq:bdy_fla1} 
    488 U = U_{e} + \frac{c}{h}\left(\eta - \eta_{e}\right), 
     506\begin{equation} 
     507  \label{eq:LBC_bdy_fla1} 
     508  U = U_{e} + \frac{c}{h}\left(\eta - \eta_{e}\right), 
    489509\end{equation} 
    490510where $U$ is the depth-mean velocity normal to the boundary and $\eta$ is the sea surface height, 
     
    495515the external depth-mean normal velocity, 
    496516plus a correction term that allows gravity waves generated internally to exit the model boundary. 
    497 Note that the sea-surface height gradient in \autoref{eq:bdy_fla1} is a spatial gradient across the model boundary, 
     517Note that the sea-surface height gradient in \autoref{eq:LBC_bdy_fla1} is a spatial gradient across the model boundary, 
    498518so that $\eta_{e}$ is defined on the $T$ points with $nbr=1$ and $\eta$ is defined on the $T$ points with $nbr=2$. 
    499519$U$ and $U_{e}$ are defined on the $U$ or $V$ points with $nbr=1$, \ie\ between the two $T$ grid points. 
     
    501521%---------------------------------------------- 
    502522\subsection{Orlanski radiation scheme} 
    503 \label{subsec:BDY_orlanski_scheme} 
     523\label{subsec:LBC_bdy_orlanski_scheme} 
    504524 
    505525The Orlanski scheme is based on the algorithm described by \citep{marchesiello.mcwilliams.ea_OM01}, hereafter MMS. 
     
    507527The adaptive Orlanski condition solves a wave plus relaxation equation at the boundary: 
    508528\begin{equation} 
    509 \frac{\partial\phi}{\partial t} + c_x \frac{\partial\phi}{\partial x} + c_y \frac{\partial\phi}{\partial y} = 
    510                                                 -\frac{1}{\tau}(\phi - \phi^{ext}) 
    511 \label{eq:wave_continuous} 
     529  \label{eq:LBC_wave_continuous} 
     530  \frac{\partial\phi}{\partial t} + c_x \frac{\partial\phi}{\partial x} + c_y \frac{\partial\phi}{\partial y} = 
     531  -\frac{1}{\tau}(\phi - \phi^{ext}) 
    512532\end{equation} 
    513533 
     
    515535velocities are diagnosed from the model fields as: 
    516536 
    517 \begin{equation} \label{eq:cx} 
    518 c_x = -\frac{\partial\phi}{\partial t}\frac{\partial\phi / \partial x}{(\partial\phi /\partial x)^2 + (\partial\phi /\partial y)^2} 
     537\begin{equation} 
     538  \label{eq:LBC_cx} 
     539  c_x = -\frac{\partial\phi}{\partial t}\frac{\partial\phi / \partial x}{(\partial\phi /\partial x)^2 + (\partial\phi /\partial y)^2} 
    519540\end{equation} 
    520541\begin{equation} 
    521 \label{eq:cy} 
    522 c_y = -\frac{\partial\phi}{\partial t}\frac{\partial\phi / \partial y}{(\partial\phi /\partial x)^2 + (\partial\phi /\partial y)^2} 
     542  \label{eq:LBC_cy} 
     543  c_y = -\frac{\partial\phi}{\partial t}\frac{\partial\phi / \partial y}{(\partial\phi /\partial x)^2 + (\partial\phi /\partial y)^2} 
    523544\end{equation} 
    524545 
    525546(As noted by MMS, this is a circular diagnosis of the phase speeds which only makes sense on a discrete grid). 
    526 Equation (\autoref{eq:wave_continuous}) is defined adaptively depending on the sign of the phase velocity normal to the boundary $c_x$. 
     547Equation (\autoref{eq:LBC_wave_continuous}) is defined adaptively depending on the sign of the phase velocity normal to the boundary $c_x$. 
    527548For $c_x$ outward, we have 
    528549 
     
    534555 
    535556\begin{equation} 
    536 \tau = \tau_{in}\,\,\,;\,\,\, c_x = c_y = 0 
    537 \label{eq:tau_in} 
     557  \label{eq:LBC_tau_in} 
     558  \tau = \tau_{in}\,\,\,;\,\,\, c_x = c_y = 0 
    538559\end{equation} 
    539560 
    540561Generally the relaxation time scale at inward propagation points (\np{rn\_time\_dmp}) is set much shorter than the time scale at outward propagation 
    541562points (\np{rn\_time\_dmp\_out}) so that the solution is constrained more strongly by the external data at inward propagation points. 
    542 See \autoref{subsec:BDY_relaxation} for detailed on the spatial shape of the scaling.\\ 
     563See \autoref{subsec:LBC_bdy_relaxation} for detailed on the spatial shape of the scaling.\\ 
    543564The ``normal propagation of oblique radiation'' or NPO approximation (called \forcode{'orlanski_npo'}) involves assuming 
    544 that $c_y$ is zero in equation (\autoref{eq:wave_continuous}), but including 
    545 this term in the denominator of equation (\autoref{eq:cx}). Both versions of the scheme are options in BDY. Equations 
    546 (\autoref{eq:wave_continuous}) - (\autoref{eq:tau_in}) correspond to equations (13) - (15) and (2) - (3) in MMS.\\ 
     565that $c_y$ is zero in equation (\autoref{eq:LBC_wave_continuous}), but including 
     566this term in the denominator of equation (\autoref{eq:LBC_cx}). Both versions of the scheme are options in BDY. Equations 
     567(\autoref{eq:LBC_wave_continuous}) - (\autoref{eq:LBC_tau_in}) correspond to equations (13) - (15) and (2) - (3) in MMS.\\ 
    547568 
    548569%---------------------------------------------- 
    549570\subsection{Relaxation at the boundary} 
    550 \label{subsec:BDY_relaxation} 
     571\label{subsec:LBC_bdy_relaxation} 
    551572 
    552573In addition to a specific boundary condition specified as \np{cn\_tra} and \np{cn\_dyn3d}, relaxation on baroclinic velocities and tracers variables are available. 
     
    564585%---------------------------------------------- 
    565586\subsection{Boundary geometry} 
    566 \label{subsec:BDY_geometry} 
     587\label{subsec:LBC_bdy_geometry} 
    567588 
    568589Each open boundary set is defined as a list of points. 
     
    615636%---------------------------------------------- 
    616637\subsection{Input boundary data files} 
    617 \label{subsec:BDY_data} 
     638\label{subsec:LBC_bdy_data} 
    618639 
    619640The data files contain the data arrays in the order in which the points are defined in the $nbi$ and $nbj$ arrays. 
     
    655676%---------------------------------------------- 
    656677\subsection{Volume correction} 
    657 \label{subsec:BDY_vol_corr} 
     678\label{subsec:LBC_bdy_vol_corr} 
    658679 
    659680There is an option to force the total volume in the regional model to be constant. 
     
    672693%---------------------------------------------- 
    673694\subsection{Tidal harmonic forcing} 
    674 \label{subsec:BDY_tides} 
     695\label{subsec:LBC_bdy_tides} 
    675696 
    676697%-----------------------------------------nambdy_tide-------------------------------------------- 
  • NEMO/trunk/doc/latex/NEMO/subfiles/chap_LDF.tex

    r11537 r11543  
    1313\newpage 
    1414 
    15 The lateral physics terms in the momentum and tracer equations have been described in \autoref{eq:PE_zdf} and 
     15The lateral physics terms in the momentum and tracer equations have been described in \autoref{eq:MB_zdf} and 
    1616their discrete formulation in \autoref{sec:TRA_ldf} and \autoref{sec:DYN_ldf}). 
    1717In this section we further discuss each lateral physics option. 
     
    2525Note that this chapter describes the standard implementation of iso-neutral tracer mixing.  
    2626Griffies's implementation, which is used if \np{ln\_traldf\_triad}\forcode{=.true.}, 
    27 is described in \autoref{apdx:triad} 
     27is described in \autoref{apdx:TRIADS} 
    2828 
    2929%-----------------------------------namtra_ldf - namdyn_ldf-------------------------------------------- 
     
    8282the cell of the quantity to be diffused. 
    8383For a tracer, this leads to the following four slopes: 
    84 $r_{1u}$, $r_{1w}$, $r_{2v}$, $r_{2w}$ (see \autoref{eq:tra_ldf_iso}), 
     84$r_{1u}$, $r_{1w}$, $r_{2v}$, $r_{2w}$ (see \autoref{eq:TRA_ldf_iso}), 
    8585while for momentum the slopes are  $r_{1t}$, $r_{1uw}$, $r_{2f}$, $r_{2uw}$ for $u$ and 
    8686$r_{1f}$, $r_{1vw}$, $r_{2t}$, $r_{2vw}$ for $v$.  
     
    9292In $s$-coordinates, geopotential mixing (\ie\ horizontal mixing) $r_1$ and $r_2$ are the slopes between 
    9393the geopotential and computational surfaces. 
    94 Their discrete formulation is found by locally solving \autoref{eq:tra_ldf_iso} when 
     94Their discrete formulation is found by locally solving \autoref{eq:TRA_ldf_iso} when 
    9595the diffusive fluxes in the three directions are set to zero and $T$ is assumed to be horizontally uniform, 
    9696\ie\ a linear function of $z_T$, the depth of a $T$-point.  
     
    9898 
    9999\begin{equation} 
    100   \label{eq:ldfslp_geo} 
     100  \label{eq:LDF_slp_geo} 
    101101  \begin{aligned} 
    102102    r_{1u} &= \frac{e_{3u}}{ \left( e_{1u}\;\overline{\overline{e_{3w}}}^{\,i+1/2,\,k} \right)} 
     
    125125Their discrete formulation is found using the fact that the diffusive fluxes of 
    126126locally referenced potential density (\ie\ $in situ$ density) vanish. 
    127 So, substituting $T$ by $\rho$ in \autoref{eq:tra_ldf_iso} and setting the diffusive fluxes in 
     127So, substituting $T$ by $\rho$ in \autoref{eq:TRA_ldf_iso} and setting the diffusive fluxes in 
    128128the three directions to zero leads to the following definition for the neutral slopes: 
    129129 
    130130\begin{equation} 
    131   \label{eq:ldfslp_iso} 
     131  \label{eq:LDF_slp_iso} 
    132132  \begin{split} 
    133133    r_{1u} &= \frac{e_{3u}}{e_{1u}}\; \frac{\delta_{i+1/2}[\rho]} 
     
    145145 
    146146%gm% rewrite this as the explanation is not very clear !!! 
    147 %In practice, \autoref{eq:ldfslp_iso} is of little help in evaluating the neutral surface slopes. Indeed, for an unsimplified equation of state, the density has a strong dependancy on pressure (here approximated as the depth), therefore applying \autoref{eq:ldfslp_iso} using the $in situ$ density, $\rho$, computed at T-points leads to a flattening of slopes as the depth increases. This is due to the strong increase of the $in situ$ density with depth.  
    148  
    149 %By definition, neutral surfaces are tangent to the local $in situ$ density \citep{mcdougall_JPO87}, therefore in \autoref{eq:ldfslp_iso}, all the derivatives have to be evaluated at the same local pressure (which in decibars is approximated by the depth in meters). 
    150  
    151 %In the $z$-coordinate, the derivative of the  \autoref{eq:ldfslp_iso} numerator is evaluated at the same depth \nocite{as what?} ($T$-level, which is the same as the $u$- and $v$-levels), so  the $in situ$ density can be used for its evaluation.  
    152  
    153 As the mixing is performed along neutral surfaces, the gradient of $\rho$ in \autoref{eq:ldfslp_iso} has to 
     147%In practice, \autoref{eq:LDF_slp_iso} is of little help in evaluating the neutral surface slopes. Indeed, for an unsimplified equation of state, the density has a strong dependancy on pressure (here approximated as the depth), therefore applying \autoref{eq:LDF_slp_iso} using the $in situ$ density, $\rho$, computed at T-points leads to a flattening of slopes as the depth increases. This is due to the strong increase of the $in situ$ density with depth.  
     148 
     149%By definition, neutral surfaces are tangent to the local $in situ$ density \citep{mcdougall_JPO87}, therefore in \autoref{eq:LDF_slp_iso}, all the derivatives have to be evaluated at the same local pressure (which in decibars is approximated by the depth in meters). 
     150 
     151%In the $z$-coordinate, the derivative of the  \autoref{eq:LDF_slp_iso} numerator is evaluated at the same depth \nocite{as what?} ($T$-level, which is the same as the $u$- and $v$-levels), so  the $in situ$ density can be used for its evaluation.  
     152 
     153As the mixing is performed along neutral surfaces, the gradient of $\rho$ in \autoref{eq:LDF_slp_iso} has to 
    154154be evaluated at the same local pressure 
    155155(which, in decibars, is approximated by the depth in meters in the model). 
    156 Therefore \autoref{eq:ldfslp_iso} cannot be used as such, 
     156Therefore \autoref{eq:LDF_slp_iso} cannot be used as such, 
    157157but further transformation is needed depending on the vertical coordinate used: 
    158158 
     
    160160 
    161161\item[$z$-coordinate with full step: ] 
    162   in \autoref{eq:ldfslp_iso} the densities appearing in the $i$ and $j$ derivatives  are taken at the same depth, 
     162  in \autoref{eq:LDF_slp_iso} the densities appearing in the $i$ and $j$ derivatives  are taken at the same depth, 
    163163  thus the $in situ$ density can be used. 
    164164  This is not the case for the vertical derivatives: $\delta_{k+1/2}[\rho]$ is replaced by $-\rho N^2/g$, 
     
    173173  in the current release of \NEMO, iso-neutral mixing is only employed for $s$-coordinates if 
    174174  the Griffies scheme is used (\np{ln\_traldf\_triad}\forcode{=.true.}; 
    175   see \autoref{apdx:triad}). 
     175  see \autoref{apdx:TRIADS}). 
    176176  In other words, iso-neutral mixing will only be accurately represented with a linear equation of state 
    177177  (\np{ln\_seos}\forcode{=.true.}). 
    178   In the case of a "true" equation of state, the evaluation of $i$ and $j$ derivatives in \autoref{eq:ldfslp_iso} 
     178  In the case of a "true" equation of state, the evaluation of $i$ and $j$ derivatives in \autoref{eq:LDF_slp_iso} 
    179179  will include a pressure dependent part, leading to the wrong evaluation of the neutral slopes. 
    180180 
     
    193193 
    194194\[ 
    195   % \label{eq:ldfslp_iso2} 
     195  % \label{eq:LDF_slp_iso2} 
    196196  \begin{split} 
    197197    r_{1u} &= \frac{e_{3u}}{e_{1u}}\; \frac 
     
    230230To overcome this problem, several techniques have been proposed in which the numerical schemes of 
    231231the ocean model are modified \citep{weaver.eby_JPO97, griffies.gnanadesikan.ea_JPO98}. 
    232 Griffies's scheme is now available in \NEMO\ if \np{ln\_traldf\_triad}\forcode{=.true.}; see \autoref{apdx:triad}. 
     232Griffies's scheme is now available in \NEMO\ if \np{ln\_traldf\_triad}\forcode{ = .true.}; see \autoref{apdx:TRIADS}. 
    233233Here, another strategy is presented \citep{lazar_phd97}: 
    234234a local filtering of the iso-neutral slopes (made on 9 grid-points) prevents the development of 
     
    280280    \includegraphics[width=\textwidth]{Fig_eiv_slp} 
    281281    \caption{ 
    282       \protect\label{fig:eiv_slp} 
     282      \protect\label{fig:LDF_eiv_slp} 
    283283      Vertical profile of the slope used for lateral mixing in the mixed layer: 
    284284      \textit{(a)} in the real ocean the slope is the iso-neutral slope in the ocean interior, 
     
    304304The iso-neutral diffusion operator on momentum is the same as the one used on tracers but 
    305305applied to each component of the velocity separately 
    306 (see \autoref{eq:dyn_ldf_iso} in section~\autoref{subsec:DYN_ldf_iso}). 
     306(see \autoref{eq:DYN_ldf_iso} in section~\autoref{subsec:DYN_ldf_iso}). 
    307307The slopes between the surface along which the diffusion operator acts and the surface of computation 
    308308($z$- or $s$-surfaces) are defined at $T$-, $f$-, and \textit{uw}- points for the $u$-component, and $T$-, $f$- and 
    309309\textit{vw}- points for the $v$-component. 
    310310They are computed from the slopes used for tracer diffusion, 
    311 \ie\ \autoref{eq:ldfslp_geo} and \autoref{eq:ldfslp_iso}: 
     311\ie\ \autoref{eq:LDF_slp_geo} and \autoref{eq:LDF_slp_iso}: 
    312312 
    313313\[ 
    314   % \label{eq:ldfslp_dyn} 
     314  % \label{eq:LDF_slp_dyn} 
    315315  \begin{aligned} 
    316316    &r_{1t}\ \ = \overline{r_{1u}}^{\,i}       &&&    r_{1f}\ \ &= \overline{r_{1u}}^{\,i+1/2} \\ 
     
    371371 
    372372\begin{equation} 
    373   \label{eq:constantah} 
     373  \label{eq:LDF_constantah} 
    374374  A_o^l = \left\{ 
    375375    \begin{aligned} 
     
    386386 
    387387In the vertically varying case, a hyperbolic variation of the lateral mixing coefficient is introduced in which 
    388 the surface value is given by \autoref{eq:constantah}, the bottom value is 1/4 of the surface value, 
     388the surface value is given by \autoref{eq:LDF_constantah}, the bottom value is 1/4 of the surface value, 
    389389and the transition takes place around z=500~m with a width of 200~m. 
    390390This profile is hard coded in module \mdl{ldfc1d\_c2d}, but can be easily modified by users. 
     
    396396the type of operator used: 
    397397\begin{equation} 
    398   \label{eq:title} 
     398  \label{eq:LDF_title} 
    399399  A_l = \left\{ 
    400400    \begin{aligned} 
     
    411411model configurations presenting large changes in grid spacing such as global ocean models. 
    412412Indeed, in such a case, a constant mixing coefficient can lead to a blow up of the model due to 
    413 large coefficient compare to the smallest grid size (see \autoref{sec:STP_forward_imp}), 
     413large coefficient compare to the smallest grid size (see \autoref{sec:TD_forward_imp}), 
    414414especially when using a bilaplacian operator. 
    415415 
     
    429429 
    430430\begin{equation} 
    431   \label{eq:flowah} 
     431  \label{eq:LDF_flowah} 
    432432  A_l = \left\{ 
    433433    \begin{aligned} 
     
    445445 
    446446\begin{equation} 
    447   \label{eq:smag1} 
     447  \label{eq:LDF_smag1} 
    448448  \begin{split} 
    449449    T_{smag}^{-1} & = \sqrt{\left( \partial_x u - \partial_y v\right)^2 + \left( \partial_y u + \partial_x v\right)^2  } \\ 
     
    455455 
    456456\begin{equation} 
    457   \label{eq:smag2} 
     457  \label{eq:LDF_smag2} 
    458458  A_{smag} = \left\{ 
    459459    \begin{aligned} 
     
    464464\end{equation} 
    465465 
    466 For stability reasons, upper and lower limits are applied on the resulting coefficient (see \autoref{sec:STP_forward_imp}) so that: 
    467 \begin{equation} 
    468   \label{eq:smag3} 
     466For stability reasons, upper and lower limits are applied on the resulting coefficient (see \autoref{sec:TD_forward_imp}) so that: 
     467\begin{equation} 
     468  \label{eq:LDF_smag3} 
    469469    \begin{aligned} 
    470470      & C_{min} \frac{1}{2}   \lvert U \rvert  e    < A_{smag} < C_{max} \frac{e^2}{   8\rdt}                 & \text{for laplacian operator } \\ 
     
    480480 
    481481(1) the momentum diffusion operator acting along model level surfaces is written in terms of curl and 
    482 divergent components of the horizontal current (see \autoref{subsec:PE_ldf}). 
     482divergent components of the horizontal current (see \autoref{subsec:MB_ldf}). 
    483483Although the eddy coefficient could be set to different values in these two terms, 
    484484this option is not currently available.  
     
    486486(2) with an horizontally varying viscosity, the quadratic integral constraints on enstrophy and on the square of 
    487487the horizontal divergence for operators acting along model-surfaces are no longer satisfied 
    488 (\autoref{sec:dynldf_properties}). 
     488(\autoref{sec:INVARIANTS_dynldf_properties}). 
    489489 
    490490% ================================================================ 
     
    527527the formulation of which depends on the slopes of iso-neutral surfaces. 
    528528Contrary to the case of iso-neutral mixing, the slopes used here are referenced to the geopotential surfaces, 
    529 \ie\ \autoref{eq:ldfslp_geo} is used in $z$-coordinates, 
    530 and the sum \autoref{eq:ldfslp_geo} + \autoref{eq:ldfslp_iso} in $s$-coordinates. 
     529\ie\ \autoref{eq:LDF_slp_geo} is used in $z$-coordinates, 
     530and the sum \autoref{eq:LDF_slp_geo} + \autoref{eq:LDF_slp_iso} in $s$-coordinates. 
    531531 
    532532If isopycnal mixing is used in the standard way, \ie\ \np{ln\_traldf\_triad}\forcode{=.false.}, the eddy induced velocity is given by:  
    533533\begin{equation} 
    534   \label{eq:ldfeiv} 
     534  \label{eq:LDF_eiv} 
    535535  \begin{split} 
    536536    u^* & = \frac{1}{e_{2u}e_{3u}}\; \delta_k \left[e_{2u} \, A_{uw}^{eiv} \; \overline{r_{1w}}^{\,i+1/2} \right]\\ 
     
    554554\colorbox{yellow}{CASE \np{nn\_aei\_ijk\_t} = 21 to be added} 
    555555 
    556 In case of setting \np{ln\_traldf\_triad}\forcode{=.true.}, a skew form of the eddy induced advective fluxes is used, which is described in \autoref{apdx:triad}. 
     556In case of setting \np{ln\_traldf\_triad}\forcode{ = .true.}, a skew form of the eddy induced advective fluxes is used, which is described in \autoref{apdx:TRIADS}. 
    557557 
    558558% ================================================================ 
  • NEMO/trunk/doc/latex/NEMO/subfiles/chap_OBS.tex

    r11435 r11543  
    691691 
    692692Examples of the weights calculated for an observation with rectangular and radial footprints are shown in 
    693 \autoref{fig:obsavgrec} and~\autoref{fig:obsavgrad}. 
     693\autoref{fig:OBS_avgrec} and~\autoref{fig:OBS_avgrad}. 
    694694 
    695695%>>>>>>>>>>>>>>>>>>>>>>>>>>>> 
     
    698698    \includegraphics[width=\textwidth]{Fig_OBS_avg_rec} 
    699699    \caption{ 
    700       \protect\label{fig:obsavgrec} 
     700      \protect\label{fig:OBS_avgrec} 
    701701      Weights associated with each model grid box (blue lines and numbers) 
    702702      for an observation at -170.5\deg{E}, 56.0\deg{N} with a rectangular footprint of 1\deg x 1\deg. 
     
    711711    \includegraphics[width=\textwidth]{Fig_OBS_avg_rad} 
    712712    \caption{ 
    713       \protect\label{fig:obsavgrad} 
     713      \protect\label{fig:OBS_avgrad} 
    714714      Weights associated with each model grid box (blue lines and numbers) 
    715715      for an observation at -170.5\deg{E}, 56.0\deg{N} with a radial footprint with diameter 1\deg. 
     
    756756          ({\phi_{}}_{\mathrm D}  \;  - \; {\phi_{}}_{\mathrm P} )] \; \widehat{\mathbf k} \\ 
    757757  \end{array} 
    758   % \label{eq:cross} 
     758  % \label{eq:OBS_cross} 
    759759\end{align*} 
    760760point in the opposite direction to the unit normal $\widehat{\mathbf k}$ 
     
    791791    \includegraphics[width=\textwidth]{Fig_ASM_obsdist_local} 
    792792    \caption{ 
    793       \protect\label{fig:obslocal} 
     793      \protect\label{fig:OBS_local} 
    794794      Example of the distribution of observations with the geographical distribution of observational data. 
    795795    } 
     
    800800This is the simplest option in which the observations are distributed according to 
    801801the domain of the grid-point parallelization. 
    802 \autoref{fig:obslocal} shows an example of the distribution of the {\em in situ} data on processors with 
     802\autoref{fig:OBS_local} shows an example of the distribution of the {\em in situ} data on processors with 
    803803a different colour for each observation on a given processor for a 4 $\times$ 2 decomposition with ORCA2. 
    804804The grid-point domain decomposition is clearly visible on the plot. 
     
    820820    \includegraphics[width=\textwidth]{Fig_ASM_obsdist_global} 
    821821    \caption{ 
    822       \protect\label{fig:obsglobal} 
     822      \protect\label{fig:OBS_global} 
    823823      Example of the distribution of observations with the round-robin distribution of observational data. 
    824824    } 
     
    830830use message passing in order to retrieve the stencil for interpolation. 
    831831The simplest distribution of the observations is to distribute them using a round-robin scheme. 
    832 \autoref{fig:obsglobal} shows the distribution of the {\em in situ} data on processors for 
     832\autoref{fig:OBS_global} shows the distribution of the {\em in situ} data on processors for 
    833833the round-robin distribution of observations with a different colour for each observation on a given processor for 
    834 a 4 $\times$ 2 decomposition with ORCA2 for the same input data as in \autoref{fig:obslocal}. 
     834a 4 $\times$ 2 decomposition with ORCA2 for the same input data as in \autoref{fig:OBS_local}. 
    835835The observations are now clearly randomly distributed on the globe. 
    836836In order to be able to perform horizontal interpolation in this case, 
     
    11181118\end{minted} 
    11191119 
    1120 \autoref{fig:obsdataplotmain} shows the main window which is launched when dataplot starts. 
     1120\autoref{fig:OBS_dataplotmain} shows the main window which is launched when dataplot starts. 
    11211121This is split into three parts. 
    11221122At the top there is a menu bar which contains a variety of drop down menus. 
     
    11541154    \includegraphics[width=\textwidth]{Fig_OBS_dataplot_main} 
    11551155    \caption{ 
    1156       \protect\label{fig:obsdataplotmain} 
     1156      \protect\label{fig:OBS_dataplotmain} 
    11571157      Main window of dataplot. 
    11581158    } 
     
    11621162 
    11631163If a profile point is clicked with the mouse button a plot of the observation and background values as 
    1164 a function of depth (\autoref{fig:obsdataplotprofile}). 
     1164a function of depth (\autoref{fig:OBS_dataplotprofile}). 
    11651165 
    11661166%>>>>>>>>>>>>>>>>>>>>>>>>>>>> 
     
    11701170    \includegraphics[width=\textwidth]{Fig_OBS_dataplot_prof} 
    11711171    \caption{ 
    1172       \protect\label{fig:obsdataplotprofile} 
     1172      \protect\label{fig:OBS_dataplotprofile} 
    11731173      Profile plot from dataplot produced by right clicking on a point in the main window. 
    11741174    } 
  • NEMO/trunk/doc/latex/NEMO/subfiles/chap_SBC.tex

    r11537 r11543  
    109109Next, the scheme for interpolation on the fly is described. 
    110110Finally, the different options that further modify the fluxes applied to the ocean are discussed. 
    111 One of these is modification by icebergs (see \autoref{sec:ICB_icebergs}), 
     111One of these is modification by icebergs (see \autoref{sec:SBC_ICB_icebergs}), 
    112112which act as drifting sources of fresh water. 
    113113Another example of modification is that due to the ice shelf melting/freezing (see \autoref{sec:SBC_isf}), 
     
    124124The surface ocean stress is the stress exerted by the wind and the sea-ice on the ocean. 
    125125It is applied in \mdl{dynzdf} module as a surface boundary condition of the computation of 
    126 the momentum vertical mixing trend (see \autoref{eq:dynzdf_sbc} in \autoref{sec:DYN_zdf}). 
     126the momentum vertical mixing trend (see \autoref{eq:DYN_zdf_sbc} in \autoref{sec:DYN_zdf}). 
    127127As such, it has to be provided as a 2D vector interpolated onto the horizontal velocity ocean mesh, 
    128128\ie\ resolved onto the model (\textbf{i},\textbf{j}) direction at $u$- and $v$-points. 
     
    135135It is applied in \mdl{trasbc} module as a surface boundary condition trend of 
    136136the first level temperature time evolution equation 
    137 (see \autoref{eq:tra_sbc} and \autoref{eq:tra_sbc_lin} in \autoref{subsec:TRA_sbc}). 
     137(see \autoref{eq:TRA_sbc} and \autoref{eq:TRA_sbc_lin} in \autoref{subsec:TRA_sbc}). 
    138138The latter is the penetrative part of the heat flux. 
    139139It is applied as a 3D trend of the temperature equation (\mdl{traqsr} module) when 
     
    177177The ocean model provides, at each time step, to the surface module (\mdl{sbcmod}) 
    178178the surface currents, temperature and salinity. 
    179 These variables are averaged over \np{nn\_fsbc} time-step (\autoref{tab:ssm}), and 
     179These variables are averaged over \np{nn\_fsbc} time-step (\autoref{tab:SBC_ssm}), and 
    180180these averaged fields are used to compute the surface fluxes at the frequency of \np{nn\_fsbc} time-steps. 
    181181 
     
    193193    \end{tabular} 
    194194    \caption{ 
    195       \protect\label{tab:ssm} 
     195      \protect\label{tab:SBC_ssm} 
    196196      Ocean variables provided by the ocean to the surface module (SBC). 
    197197      The variable are averaged over \np{nn\_fsbc} time-step, 
     
    264264  This stem will be completed automatically by the model, with the addition of a '.nc' at its end and 
    265265  by date information and possibly a prefix (when using AGRIF). 
    266   \autoref{tab:fldread} provides the resulting file name in all possible cases according to 
     266  \autoref{tab:SBC_fldread} provides the resulting file name in all possible cases according to 
    267267  whether it is a climatological file or not, and to the open/close frequency (see below for definition). 
    268268 
     
    278278    \end{center} 
    279279    \caption{ 
    280       \protect\label{tab:fldread} 
     280      \protect\label{tab:SBC_fldread} 
    281281      naming nomenclature for climatological or interannual input file(s), as a function of the open/close frequency. 
    282282      The stem name is assumed to be 'fn'. 
     
    515515% ------------------------------------------------------------------------------------------------------------- 
    516516\subsection{Standalone surface boundary condition scheme (SAS)} 
    517 \label{subsec:SAS} 
     517\label{subsec:SBC_SAS} 
    518518 
    519519%---------------------------------------namsbc_sas-------------------------------------------------- 
     
    649649%--------------------------------------------------TABLE-------------------------------------------------- 
    650650\begin{table}[htbp] 
    651   \label{tab:BULK} 
     651  \label{tab:SBC_BULK} 
    652652  \begin{center} 
    653653    \begin{tabular}{|l|c|c|c|} 
     
    852852The tidal forcing, generated by the gravity forces of the Earth-Moon and Earth-Sun sytems, 
    853853is activated if \np{ln\_tide} and \np{ln\_tide\_pot} are both set to \forcode{.true.} in \nam{\_tide}. 
    854 This translates as an additional barotropic force in the momentum equations \ref{eq:PE_dyn} such that: 
    855 \[ 
    856   % \label{eq:PE_dyn_tides} 
     854This translates as an additional barotropic force in the momentum \autoref{eq:MB_PE_dyn} such that: 
     855\[ 
     856  % \label{eq:SBC_PE_dyn_tides} 
    857857  \frac{\partial {\mathrm {\mathbf U}}_h }{\partial t}= ... 
    858858  +g\nabla (\Pi_{eq} + \Pi_{sal}) 
     
    895895%        River runoffs 
    896896% ================================================================ 
    897 \section[River runoffs (\textit{sbcrnf.F90})] 
    898 {River runoffs (\protect\mdl{sbcrnf})} 
     897\section[River runoffs (\textit{sbcrnf.F90})]{River runoffs (\protect\mdl{sbcrnf})} 
    899898\label{sec:SBC_rnf} 
    900899%------------------------------------------namsbc_rnf---------------------------------------------------- 
     
    10221021%        Ice shelf melting 
    10231022% ================================================================ 
    1024 \section[Ice shelf melting (\textit{sbcisf.F90})] 
    1025 {Ice shelf melting (\protect\mdl{sbcisf})} 
     1023\section[Ice shelf melting (\textit{sbcisf.F90})]{Ice shelf melting (\protect\mdl{sbcisf})} 
    10261024\label{sec:SBC_isf} 
    10271025%------------------------------------------namsbc_isf---------------------------------------------------- 
     
    10661064     The salt and heat exchange coefficients are constant and defined by \np{rn\_gammas0} and \np{rn\_gammat0}. 
    10671065\[ 
    1068   % \label{eq:sbc_isf_gamma_iso} 
     1066  % \label{eq:SBC_isf_gamma_iso} 
    10691067\gamma^{T} = \np{rn\_gammat0} 
    10701068\] 
     
    12131211% ================================================================ 
    12141212\section{Handling of icebergs (ICB)} 
    1215 \label{sec:ICB_icebergs} 
     1213\label{sec:SBC_ICB_icebergs} 
    12161214%------------------------------------------namberg---------------------------------------------------- 
    12171215 
     
    12821280%        Interactions with waves (sbcwave.F90, ln_wave) 
    12831281% ============================================================================================================= 
    1284 \section[Interactions with waves (\textit{sbcwave.F90}, \texttt{ln\_wave})] 
    1285 {Interactions with waves (\protect\mdl{sbcwave}, \protect\np{ln\_wave})} 
     1282\section[Interactions with waves (\textit{sbcwave.F90}, \texttt{ln\_wave})]{Interactions with waves (\protect\mdl{sbcwave}, \protect\np{ln\_wave})} 
    12861283\label{sec:SBC_wave} 
    12871284%------------------------------------------namsbc_wave-------------------------------------------------------- 
     
    13141311 
    13151312% ---------------------------------------------------------------- 
    1316 \subsection[Neutral drag coefficient from wave model (\texttt{ln\_cdgw})] 
    1317 {Neutral drag coefficient from wave model (\protect\np{ln\_cdgw})} 
     1313\subsection[Neutral drag coefficient from wave model (\texttt{ln\_cdgw})]{Neutral drag coefficient from wave model (\protect\np{ln\_cdgw})} 
    13181314\label{subsec:SBC_wave_cdgw} 
    13191315 
     
    13281324% 3D Stokes Drift (ln_sdw, nn_sdrift) 
    13291325% ---------------------------------------------------------------- 
    1330 \subsection[3D Stokes Drift (\texttt{ln\_sdw}, \texttt{nn\_sdrift})] 
    1331 {3D Stokes Drift (\protect\np{ln\_sdw, nn\_sdrift})} 
     1326\subsection[3D Stokes Drift (\texttt{ln\_sdw}, \texttt{nn\_sdrift})]{3D Stokes Drift (\protect\np{ln\_sdw, nn\_sdrift})} 
    13321327\label{subsec:SBC_wave_sdw} 
    13331328 
     
    13431338 
    13441339\[ 
    1345   % \label{eq:sbc_wave_sdw} 
     1340  % \label{eq:SBC_wave_sdw} 
    13461341  \mathbf{U}_{st} = \frac{16{\pi^3}} {g} 
    13471342  \int_0^\infty \int_{-\pi}^{\pi} (cos{\theta},sin{\theta}) {f^3} 
     
    13681363 
    13691364\[ 
    1370   % \label{eq:sbc_wave_sdw_0a} 
     1365  % \label{eq:SBC_wave_sdw_0a} 
    13711366  \mathbf{U}_{st} \cong \mathbf{U}_{st |_{z=0}} \frac{\mathrm{e}^{-2k_ez}} {1-8k_ez} 
    13721367\] 
     
    13751370 
    13761371\[ 
    1377   % \label{eq:sbc_wave_sdw_0b} 
     1372  % \label{eq:SBC_wave_sdw_0b} 
    13781373  k_e = \frac{|\mathbf{U}_{\left.st\right|_{z=0}}|} {|T_{st}|} 
    13791374  \quad \text{and }\ 
     
    13881383 
    13891384\[ 
    1390   % \label{eq:sbc_wave_sdw_1} 
     1385  % \label{eq:SBC_wave_sdw_1} 
    13911386  \mathbf{U}_{st} \cong \mathbf{U}_{st |_{z=0}} \Big[exp(2k_pz)-\beta \sqrt{-2 \pi k_pz} 
    13921387  \textit{ erf } \Big(\sqrt{-2 k_pz}\Big)\Big] 
     
    14041399 
    14051400\[ 
    1406   % \label{eq:sbc_wave_eta_sdw} 
     1401  % \label{eq:SBC_wave_eta_sdw} 
    14071402  \frac{\partial{\eta}}{\partial{t}} = 
    14081403  -\nabla_h \int_{-H}^{\eta} (\mathbf{U} + \mathbf{U}_{st}) dz 
     
    14161411 
    14171412\[ 
    1418   % \label{eq:sbc_wave_tra_sdw} 
     1413  % \label{eq:SBC_wave_tra_sdw} 
    14191414  \frac{\partial{c}}{\partial{t}} = 
    14201415  - (\mathbf{U} + \mathbf{U}_{st}) \cdot \nabla{c} 
     
    14251420% Stokes-Coriolis term (ln_stcor) 
    14261421% ---------------------------------------------------------------- 
    1427 \subsection[Stokes-Coriolis term (\texttt{ln\_stcor})] 
    1428 {Stokes-Coriolis term (\protect\np{ln\_stcor})} 
     1422\subsection[Stokes-Coriolis term (\texttt{ln\_stcor})]{Stokes-Coriolis term (\protect\np{ln\_stcor})} 
    14291423\label{subsec:SBC_wave_stcor} 
    14301424 
     
    14401434% Waves modified stress (ln_tauwoc, ln_tauw) 
    14411435% ---------------------------------------------------------------- 
    1442 \subsection[Wave modified stress (\texttt{ln\_tauwoc}, \texttt{ln\_tauw})] 
    1443 {Wave modified sress (\protect\np{ln\_tauwoc, ln\_tauw})} 
     1436\subsection[Wave modified stress (\texttt{ln\_tauwoc}, \texttt{ln\_tauw})]{Wave modified sress (\protect\np{ln\_tauwoc, ln\_tauw})} 
    14441437\label{subsec:SBC_wave_tauw} 
    14451438 
     
    14531446 
    14541447\[ 
    1455   % \label{eq:sbc_wave_tauoc} 
     1448  % \label{eq:SBC_wave_tauoc} 
    14561449  \tau_{oc,a} = \tau_a - \tau_w 
    14571450\] 
     
    14611454 
    14621455\[ 
    1463   % \label{eq:sbc_wave_tauw} 
     1456  % \label{eq:SBC_wave_tauw} 
    14641457  \tau_w = \rho g \int {\frac{dk}{c_p} (S_{in}+S_{nl}+S_{diss})} 
    14651458\] 
     
    14901483%        Diurnal cycle 
    14911484% ------------------------------------------------------------------------------------------------------------- 
    1492 \subsection[Diurnal cycle (\textit{sbcdcy.F90})] 
    1493 {Diurnal cycle (\protect\mdl{sbcdcy})} 
     1485\subsection[Diurnal cycle (\textit{sbcdcy.F90})]{Diurnal cycle (\protect\mdl{sbcdcy})} 
    14941486\label{subsec:SBC_dcy} 
    14951487%------------------------------------------namsbc------------------------------------------------------------- 
     
    15771569%        Surface restoring to observed SST and/or SSS 
    15781570% ------------------------------------------------------------------------------------------------------------- 
    1579 \subsection[Surface restoring to observed SST and/or SSS (\textit{sbcssr.F90})] 
    1580 {Surface restoring to observed SST and/or SSS (\protect\mdl{sbcssr})} 
     1571\subsection[Surface restoring to observed SST and/or SSS (\textit{sbcssr.F90})]{Surface restoring to observed SST and/or SSS (\protect\mdl{sbcssr})} 
    15811572\label{subsec:SBC_ssr} 
    15821573%------------------------------------------namsbc_ssr---------------------------------------------------- 
     
    15891580a feedback term \emph{must} be added to the surface heat flux $Q_{ns}^o$: 
    15901581\[ 
    1591   % \label{eq:sbc_dmp_q} 
     1582  % \label{eq:SBC_dmp_q} 
    15921583  Q_{ns} = Q_{ns}^o + \frac{dQ}{dT} \left( \left. T \right|_{k=1} - SST_{Obs} \right) 
    15931584\] 
     
    16021593 
    16031594\begin{equation} 
    1604   \label{eq:sbc_dmp_emp} 
     1595  \label{eq:SBC_dmp_emp} 
    16051596  \textit{emp} = \textit{emp}_o + \gamma_s^{-1} e_{3t}  \frac{  \left(\left.S\right|_{k=1}-SSS_{Obs}\right)} 
    16061597  {\left.S\right|_{k=1}} 
     
    16131604$\left.S\right|_{k=1}$ is the model surface layer salinity and 
    16141605$\gamma_s$ is a negative feedback coefficient which is provided as a namelist parameter. 
    1615 Unlike heat flux, there is no physical justification for the feedback term in \autoref{eq:sbc_dmp_emp} as 
     1606Unlike heat flux, there is no physical justification for the feedback term in \autoref{eq:SBC_dmp_emp} as 
    16161607the atmosphere does not care about ocean surface salinity \citep{madec.delecluse_IWN97}. 
    16171608The SSS restoring term should be viewed as a flux correction on freshwater fluxes to 
     
    16631654%        CICE-ocean Interface 
    16641655% ------------------------------------------------------------------------------------------------------------- 
    1665 \subsection[Interface to CICE (\textit{sbcice\_cice.F90})] 
    1666 {Interface to CICE (\protect\mdl{sbcice\_cice})} 
     1656\subsection[Interface to CICE (\textit{sbcice\_cice.F90})]{Interface to CICE (\protect\mdl{sbcice\_cice})} 
    16671657\label{subsec:SBC_cice} 
    16681658 
     
    16981688%        Freshwater budget control 
    16991689% ------------------------------------------------------------------------------------------------------------- 
    1700 \subsection[Freshwater budget control (\textit{sbcfwb.F90})] 
    1701 {Freshwater budget control (\protect\mdl{sbcfwb})} 
     1690\subsection[Freshwater budget control (\textit{sbcfwb.F90})]{Freshwater budget control (\protect\mdl{sbcfwb})} 
    17021691\label{subsec:SBC_fwb} 
    17031692 
  • NEMO/trunk/doc/latex/NEMO/subfiles/chap_TRA.tex

    r11537 r11543  
    7373Its discrete expression is given by : 
    7474\begin{equation} 
    75   \label{eq:tra_adv} 
     75  \label{eq:TRA_adv} 
    7676  ADV_\tau = - \frac{1}{b_t} \Big(   \delta_i [ e_{2u} \, e_{3u} \; u \; \tau_u] 
    7777                                   + \delta_j [ e_{1v} \, e_{3v} \; v \; \tau_v] \Big) 
     
    7979\end{equation} 
    8080where $\tau$ is either T or S, and $b_t = e_{1t} \, e_{2t} \, e_{3t}$ is the volume of $T$-cells. 
    81 The flux form in \autoref{eq:tra_adv} implicitly requires the use of the continuity equation. 
     81The flux form in \autoref{eq:TRA_adv} implicitly requires the use of the continuity equation. 
    8282Indeed, it is obtained by using the following equality: $\nabla \cdot (\vect U \, T) = \vect U \cdot \nabla T$ which 
    8383results from the use of the continuity equation, $\partial_t e_3 + e_3 \; \nabla \cdot \vect U = 0$ 
     
    8686it is consistent with the continuity equation in order to enforce the conservation properties of 
    8787the continuous equations. 
    88 In other words, by setting $\tau=1$ in (\autoref{eq:tra_adv}) we recover the discrete form of 
     88In other words, by setting $\tau = 1$ in (\autoref{eq:TRA_adv}) we recover the discrete form of 
    8989the continuity equation which is used to calculate the vertical velocity. 
    9090%>>>>>>>>>>>>>>>>>>>>>>>>>>>> 
     
    9393    \includegraphics[width=\textwidth]{Fig_adv_scheme} 
    9494    \caption{ 
    95       \protect\label{fig:adv_scheme} 
     95      \protect\label{fig:TRA_adv_scheme} 
    9696      Schematic representation of some ways used to evaluate the tracer value at $u$-point and 
    9797      the amount of tracer exchanged between two neighbouring grid points. 
     
    112112The key difference between the advection schemes available in \NEMO\ is the choice made in space and 
    113113time interpolation to define the value of the tracer at the velocity points 
    114 (\autoref{fig:adv_scheme}). 
     114(\autoref{fig:TRA_adv_scheme}). 
    115115 
    116116Along solid lateral and bottom boundaries a zero tracer flux is automatically specified, 
     
    139139two quantities that are not correlated \citep{roullet.madec_JGR00, griffies.pacanowski.ea_MWR01, campin.adcroft.ea_OM04}. 
    140140 
    141 The velocity field that appears in (\autoref{eq:tra_adv} is 
     141The velocity field that appears in (\autoref{eq:TRA_adv} is 
    142142the centred (\textit{now}) \textit{effective} ocean velocity, \ie\ the \textit{eulerian} velocity 
    143143(see \autoref{chap:DYN}) plus the eddy induced velocity (\textit{eiv}) and/or 
     
    199199For example, in the $i$-direction : 
    200200\begin{equation} 
    201   \label{eq:tra_adv_cen2} 
     201  \label{eq:TRA_adv_cen2} 
    202202  \tau_u^{cen2} = \overline T ^{i + 1/2} 
    203203\end{equation} 
     
    208208produce a sensible solution. 
    209209The associated time-stepping is performed using a leapfrog scheme in conjunction with an Asselin time-filter, 
    210 so $T$ in (\autoref{eq:tra_adv_cen2}) is the \textit{now} tracer value. 
     210so $T$ in (\autoref{eq:TRA_adv_cen2}) is the \textit{now} tracer value. 
    211211 
    212212Note that using the CEN2, the overall tracer advection is of second order accuracy since 
    213 both (\autoref{eq:tra_adv}) and (\autoref{eq:tra_adv_cen2}) have this order of accuracy. 
     213both (\autoref{eq:TRA_adv}) and (\autoref{eq:TRA_adv_cen2}) have this order of accuracy. 
    214214 
    215215%        4nd order centred scheme 
     
    219219For example, in the $i$-direction: 
    220220\begin{equation} 
    221   \label{eq:tra_adv_cen4} 
     221  \label{eq:TRA_adv_cen4} 
    222222  \tau_u^{cen4} = \overline{T - \frac{1}{6} \, \delta_i \Big[ \delta_{i + 1/2}[T] \, \Big]}^{\,i + 1/2} 
    223223\end{equation} 
     
    229229Strictly speaking, the CEN4 scheme is not a $4^{th}$ order advection scheme but 
    230230a $4^{th}$ order evaluation of advective fluxes, 
    231 since the divergence of advective fluxes \autoref{eq:tra_adv} is kept at $2^{nd}$ order. 
     231since the divergence of advective fluxes \autoref{eq:TRA_adv} is kept at $2^{nd}$ order. 
    232232The expression \textit{$4^{th}$ order scheme} used in oceanographic literature is usually associated with 
    233233the scheme presented here. 
     
    240240Furthermore, it must be used in conjunction with an explicit diffusion operator to produce a sensible solution. 
    241241As in CEN2 case, the time-stepping is performed using a leapfrog scheme in conjunction with an Asselin time-filter, 
    242 so $T$ in (\autoref{eq:tra_adv_cen4}) is the \textit{now} tracer. 
     242so $T$ in (\autoref{eq:TRA_adv_cen4}) is the \textit{now} tracer. 
    243243 
    244244At a $T$-grid cell adjacent to a boundary (coastline, bottom and surface), 
     
    265265For example, in the $i$-direction : 
    266266\begin{equation} 
    267   \label{eq:tra_adv_fct} 
     267  \label{eq:TRA_adv_fct} 
    268268  \begin{split} 
    269269    \tau_u^{ups} &= 
     
    287287 
    288288 
    289 For stability reasons (see \autoref{chap:STP}), 
    290 $\tau_u^{cen}$ is evaluated in (\autoref{eq:tra_adv_fct}) using the \textit{now} tracer while 
     289For stability reasons (see \autoref{chap:TD}), 
     290$\tau_u^{cen}$ is evaluated in (\autoref{eq:TRA_adv_fct}) using the \textit{now} tracer while 
    291291$\tau_u^{ups}$ is evaluated using the \textit{before} tracer. 
    292292In other words, the advective part of the scheme is time stepped with a leap-frog scheme 
     
    305305MUSCL has been first implemented in \NEMO\ by \citet{levy.estublier.ea_GRL01}. 
    306306In its formulation, the tracer at velocity points is evaluated assuming a linear tracer variation between 
    307 two $T$-points (\autoref{fig:adv_scheme}). 
     307two $T$-points (\autoref{fig:TRA_adv_scheme}). 
    308308For example, in the $i$-direction : 
    309309\begin{equation} 
    310   % \label{eq:tra_adv_mus} 
     310  % \label{eq:TRA_adv_mus} 
    311311  \tau_u^{mus} = \lt\{ 
    312312  \begin{split} 
     
    345345For example, in the $i$-direction: 
    346346\begin{equation} 
    347   \label{eq:tra_adv_ubs} 
     347  \label{eq:TRA_adv_ubs} 
    348348  \tau_u^{ubs} = \overline T ^{i + 1/2} - \frac{1}{6} 
    349349    \begin{cases} 
     
    369369(\np{nn\_ubs\_v}\forcode{=2 or 4}). 
    370370 
    371 For stability reasons (see \autoref{chap:STP}), the first term  in \autoref{eq:tra_adv_ubs} 
     371For stability reasons (see \autoref{chap:TD}), the first term  in \autoref{eq:TRA_adv_ubs} 
    372372(which corresponds to a second order centred scheme) 
    373373is evaluated using the \textit{now} tracer (centred in time) while the second term 
     
    376376This choice is discussed by \citet{webb.de-cuevas.ea_JAOT98} in the context of the QUICK advection scheme. 
    377377UBS and QUICK schemes only differ by one coefficient. 
    378 Replacing 1/6 with 1/8 in \autoref{eq:tra_adv_ubs} leads to the QUICK advection scheme \citep{webb.de-cuevas.ea_JAOT98}. 
     378Replacing 1/6 with 1/8 in \autoref{eq:TRA_adv_ubs} leads to the QUICK advection scheme \citep{webb.de-cuevas.ea_JAOT98}. 
    379379This option is not available through a namelist parameter, since the 1/6 coefficient is hard coded. 
    380380Nevertheless it is quite easy to make the substitution in the \mdl{traadv\_ubs} module and obtain a QUICK scheme. 
    381381 
    382 Note that it is straightforward to rewrite \autoref{eq:tra_adv_ubs} as follows: 
     382Note that it is straightforward to rewrite \autoref{eq:TRA_adv_ubs} as follows: 
    383383\begin{gather} 
    384   \label{eq:traadv_ubs2} 
     384  \label{eq:TRA_adv_ubs2} 
    385385  \tau_u^{ubs} = \tau_u^{cen4} + \frac{1}{12} 
    386386    \begin{cases} 
     
    389389    \end{cases} 
    390390  \intertext{or equivalently} 
    391   % \label{eq:traadv_ubs2b} 
     391  % \label{eq:TRA_adv_ubs2b} 
    392392  u_{i + 1/2} \ \tau_u^{ubs} = u_{i + 1/2} \, \overline{T - \frac{1}{6} \, \delta_i \Big[ \delta_{i + 1/2}[T] \Big]}^{\,i + 1/2} 
    393393                             - \frac{1}{2} |u|_{i + 1/2} \, \frac{1}{6} \, \delta_{i + 1/2} [\tau"_i] \nonumber 
    394394\end{gather} 
    395395 
    396 \autoref{eq:traadv_ubs2} has several advantages. 
     396\autoref{eq:TRA_adv_ubs2} has several advantages. 
    397397Firstly, it clearly reveals that the UBS scheme is based on the fourth order scheme to which 
    398398an upstream-biased diffusion term is added. 
    399399Secondly, this emphasises that the $4^{th}$ order part (as well as the $2^{nd}$ order part as stated above) has to 
    400 be evaluated at the \textit{now} time step using \autoref{eq:tra_adv_ubs}. 
     400be evaluated at the \textit{now} time step using \autoref{eq:TRA_adv_ubs}. 
    401401Thirdly, the diffusion term is in fact a biharmonic operator with an eddy coefficient which 
    402402is simply proportional to the velocity: $A_u^{lm} = \frac{1}{12} \, {e_{1u}}^3 \, |u|$. 
    403 Note the current version of \NEMO\ uses the computationally more efficient formulation \autoref{eq:tra_adv_ubs}. 
     403Note the current version of \NEMO\ uses the computationally more efficient formulation \autoref{eq:TRA_adv_ubs}. 
    404404 
    405405% ------------------------------------------------------------------------------------------------------------- 
     
    452452\ie\ the tracers appearing in its expression are the \textit{before} tracers in time, 
    453453except for the pure vertical component that appears when a rotation tensor is used. 
    454 This latter component is solved implicitly together with the vertical diffusion term (see \autoref{chap:STP}). 
     454This latter component is solved implicitly together with the vertical diffusion term (see \autoref{chap:TD}). 
    455455When \np{ln\_traldf\_msc}\forcode{=.true.}, a Method of Stabilizing Correction is used in which 
    456456the pure vertical component is split into an explicit and an implicit part \citep{lemarie.debreu.ea_OM12}. 
     
    527527The laplacian diffusion operator acting along the model (\textit{i,j})-surfaces is given by: 
    528528\begin{equation} 
    529   \label{eq:tra_ldf_lap} 
     529  \label{eq:TRA_ldf_lap} 
    530530  D_t^{lT} = \frac{1}{b_t} \Bigg(   \delta_{i} \lt[ A_u^{lT} \; \frac{e_{2u} \, e_{3u}}{e_{1u}} \; \delta_{i + 1/2} [T] \rt] 
    531531                                  + \delta_{j} \lt[ A_v^{lT} \; \frac{e_{1v} \, e_{3v}}{e_{2v}} \; \delta_{j + 1/2} [T] \rt] \Bigg) 
     
    547547Note that in the partial step $z$-coordinate (\np{ln\_zps}\forcode{=.true.}), 
    548548tracers in horizontally adjacent cells are located at different depths in the vicinity of the bottom. 
    549 In this case, horizontal derivatives in (\autoref{eq:tra_ldf_lap}) at the bottom level require a specific treatment. 
     549In this case, horizontal derivatives in (\autoref{eq:TRA_ldf_lap}) at the bottom level require a specific treatment. 
    550550They are calculated in the \mdl{zpshde} module, described in \autoref{sec:TRA_zpshde}. 
    551551 
     
    561561{Standard rotated (bi-)laplacian operator (\protect\mdl{traldf\_iso})} 
    562562\label{subsec:TRA_ldf_iso} 
    563 The general form of the second order lateral tracer subgrid scale physics (\autoref{eq:PE_zdf}) 
     563The general form of the second order lateral tracer subgrid scale physics (\autoref{eq:MB_zdf}) 
    564564takes the following semi -discrete space form in $z$- and $s$-coordinates: 
    565565\begin{equation} 
    566   \label{eq:tra_ldf_iso} 
     566  \label{eq:TRA_ldf_iso} 
    567567  \begin{split} 
    568568    D_T^{lT} = \frac{1}{b_t} \Bigg[ \quad &\delta_i A_u^{lT} \lt( \frac{e_{2u} e_{3u}}{e_{1u}}                      \, \delta_{i + 1/2} [T] 
     
    585585the mask technique (see \autoref{sec:LBC_coast}). 
    586586 
    587 The operator in \autoref{eq:tra_ldf_iso} involves both lateral and vertical derivatives. 
     587The operator in \autoref{eq:TRA_ldf_iso} involves both lateral and vertical derivatives. 
    588588For numerical stability, the vertical second derivative must be solved using the same implicit time scheme as that 
    589589used in the vertical physics (see \autoref{sec:TRA_zdf}). 
     
    597597 
    598598Note that in the partial step $z$-coordinate (\np{ln\_zps}\forcode{=.true.}), 
    599 the horizontal derivatives at the bottom level in \autoref{eq:tra_ldf_iso} require a specific treatment. 
     599the horizontal derivatives at the bottom level in \autoref{eq:TRA_ldf_iso} require a specific treatment. 
    600600They are calculated in module zpshde, described in \autoref{sec:TRA_zpshde}. 
    601601 
     
    608608An alternative scheme developed by \cite{griffies.gnanadesikan.ea_JPO98} which ensures tracer variance decreases 
    609609is also available in \NEMO\ (\np{ln\_traldf\_triad}\forcode{=.true.}). 
    610 A complete description of the algorithm is given in \autoref{apdx:triad}. 
    611  
    612 The lateral fourth order bilaplacian operator on tracers is obtained by applying (\autoref{eq:tra_ldf_lap}) twice. 
     610A complete description of the algorithm is given in \autoref{apdx:TRIADS}. 
     611 
     612The lateral fourth order bilaplacian operator on tracers is obtained by applying (\autoref{eq:TRA_ldf_lap}) twice. 
    613613The operator requires an additional assumption on boundary conditions: 
    614614both first and third derivative terms normal to the coast are set to zero. 
    615615 
    616 The lateral fourth order operator formulation on tracers is obtained by applying (\autoref{eq:tra_ldf_iso}) twice. 
     616The lateral fourth order operator formulation on tracers is obtained by applying (\autoref{eq:TRA_ldf_iso}) twice. 
    617617It requires an additional assumption on boundary conditions: 
    618618first and third derivative terms normal to the coast, 
     
    646646The formulation of the vertical subgrid scale tracer physics is the same for all the vertical coordinates, 
    647647and is based on a laplacian operator. 
    648 The vertical diffusion operator given by (\autoref{eq:PE_zdf}) takes the following semi -discrete space form: 
     648The vertical diffusion operator given by (\autoref{eq:MB_zdf}) takes the following semi -discrete space form: 
    649649\begin{gather*} 
    650   % \label{eq:tra_zdf} 
     650  % \label{eq:TRA_zdf} 
    651651    D^{vT}_T = \frac{1}{e_{3t}} \, \delta_k \lt[ \, \frac{A^{vT}_w}{e_{3w}} \delta_{k + 1/2}[T] \, \rt] \\ 
    652652    D^{vS}_T = \frac{1}{e_{3t}} \; \delta_k \lt[ \, \frac{A^{vS}_w}{e_{3w}} \delta_{k + 1/2}[S] \, \rt] 
     
    659659Furthermore, when iso-neutral mixing is used, both mixing coefficients are increased by 
    660660$\frac{e_{1w} e_{2w}}{e_{3w} }({r_{1w}^2 + r_{2w}^2})$ to account for the vertical second derivative of 
    661 \autoref{eq:tra_ldf_iso}. 
     661\autoref{eq:TRA_ldf_iso}. 
    662662 
    663663At the surface and bottom boundaries, the turbulent fluxes of heat and salt must be specified. 
     
    721721The surface boundary condition on temperature and salinity is applied as follows: 
    722722\begin{equation} 
    723   \label{eq:tra_sbc} 
     723  \label{eq:TRA_sbc} 
    724724  \begin{alignedat}{2} 
    725725    F^T &= \frac{1}{C_p} &\frac{1}{\rho_o \lt. e_{3t} \rt|_{k = 1}} &\overline{Q_{ns}      }^t \\ 
     
    729729where $\overline x^t$ means that $x$ is averaged over two consecutive time steps 
    730730($t - \rdt / 2$ and $t + \rdt / 2$). 
    731 Such time averaging prevents the divergence of odd and even time step (see \autoref{chap:STP}). 
     731Such time averaging prevents the divergence of odd and even time step (see \autoref{chap:TD}). 
    732732 
    733733In the linear free surface case (\np{ln\_linssh}\forcode{=.true.}), an additional term has to be added on 
     
    738738The resulting surface boundary condition is applied as follows: 
    739739\begin{equation} 
    740   \label{eq:tra_sbc_lin} 
     740  \label{eq:TRA_sbc_lin} 
    741741  \begin{alignedat}{2} 
    742742    F^T &= \frac{1}{C_p} &\frac{1}{\rho_o \lt. e_{3t} \rt|_{k = 1}} 
     
    749749In the linear free surface case, there is a small imbalance. 
    750750The imbalance is larger than the imbalance associated with the Asselin time filter \citep{leclair.madec_OM09}. 
    751 This is the reason why the modified filter is not applied in the linear free surface case (see \autoref{chap:STP}). 
     751This is the reason why the modified filter is not applied in the linear free surface case (see \autoref{chap:TD}). 
    752752 
    753753% ------------------------------------------------------------------------------------------------------------- 
     
    766766the solar radiation penetrates the top few tens of meters of the ocean. 
    767767If it is not used (\np{ln\_traqsr}\forcode{=.false.}) all the heat flux is absorbed in the first ocean level. 
    768 Thus, in the former case a term is added to the time evolution equation of temperature \autoref{eq:PE_tra_T} and 
     768Thus, in the former case a term is added to the time evolution equation of temperature \autoref{eq:MB_PE_tra_T} and 
    769769the surface boundary condition is modified to take into account only the non-penetrative part of the surface 
    770770heat flux: 
    771771\begin{equation} 
    772   \label{eq:PE_qsr} 
     772  \label{eq:TRA_PE_qsr} 
    773773  \begin{gathered} 
    774774    \pd[T]{t} = \ldots + \frac{1}{\rho_o \, C_p \, e_3} \; \pd[I]{k} \\ 
     
    778778where $Q_{sr}$ is the penetrative part of the surface heat flux (\ie\ the shortwave radiation) and 
    779779$I$ is the downward irradiance ($\lt. I \rt|_{z = \eta} = Q_{sr}$). 
    780 The additional term in \autoref{eq:PE_qsr} is discretized as follows: 
    781 \begin{equation} 
    782   \label{eq:tra_qsr} 
     780The additional term in \autoref{eq:TRA_PE_qsr} is discretized as follows: 
     781\begin{equation} 
     782  \label{eq:TRA_qsr} 
    783783  \frac{1}{\rho_o \, C_p \, e_3} \, \pd[I]{k} \equiv \frac{1}{\rho_o \, C_p \, e_{3t}} \delta_k [I_w] 
    784784\end{equation} 
     
    798798leading to the following expression \citep{paulson.simpson_JPO77}: 
    799799\[ 
    800   % \label{eq:traqsr_iradiance} 
     800  % \label{eq:TRA_qsr_iradiance} 
    801801  I(z) = Q_{sr} \lt[ Re^{- z / \xi_0} + (1 - R) e^{- z / \xi_1} \rt] 
    802802\] 
     
    807807 
    808808Such assumptions have been shown to provide a very crude and simplistic representation of 
    809 observed light penetration profiles (\cite{morel_JGR88}, see also \autoref{fig:traqsr_irradiance}). 
     809observed light penetration profiles (\cite{morel_JGR88}, see also \autoref{fig:TRA_qsr_irradiance}). 
    810810Light absorption in the ocean depends on particle concentration and is spectrally selective. 
    811811\cite{morel_JGR88} has shown that an accurate representation of light penetration can be provided by 
     
    817817the full spectral model of \cite{morel_JGR88} (as modified by \cite{morel.maritorena_JGR01}), 
    818818assuming the same power-law relationship. 
    819 As shown in \autoref{fig:traqsr_irradiance}, this formulation, called RGB (Red-Green-Blue), 
     819As shown in \autoref{fig:TRA_qsr_irradiance}, this formulation, called RGB (Red-Green-Blue), 
    820820reproduces quite closely the light penetration profiles predicted by the full spectal model, 
    821821but with much greater computational efficiency. 
     
    843843\end{description} 
    844844 
    845 The trend in \autoref{eq:tra_qsr} associated with the penetration of the solar radiation is added to 
     845The trend in \autoref{eq:TRA_qsr} associated with the penetration of the solar radiation is added to 
    846846the temperature trend, and the surface heat flux is modified in routine \mdl{traqsr}. 
    847847 
     
    860860    \includegraphics[width=\textwidth]{Fig_TRA_Irradiance} 
    861861    \caption{ 
    862       \protect\label{fig:traqsr_irradiance} 
     862      \protect\label{fig:TRA_qsr_irradiance} 
    863863      Penetration profile of the downward solar irradiance calculated by four models. 
    864864      Two waveband chlorophyll-independent formulation (blue), 
     
    888888    \includegraphics[width=\textwidth]{Fig_TRA_geoth} 
    889889    \caption{ 
    890       \protect\label{fig:geothermal} 
     890      \protect\label{fig:TRA_geothermal} 
    891891      Geothermal Heat flux (in $mW.m^{-2}$) used by \cite{emile-geay.madec_OS09}. 
    892892      It is inferred from the age of the sea floor and the formulae of \citet{stein.stein_N92}. 
     
    910910the \np{rn\_geoflx\_cst}, which is also a namelist parameter. 
    911911When \np{nn\_geoflx} is set to 2, a spatially varying geothermal heat flux is introduced which is provided in 
    912 the \ifile{geothermal\_heating} NetCDF file (\autoref{fig:geothermal}) \citep{emile-geay.madec_OS09}. 
     912the \ifile{geothermal\_heating} NetCDF file (\autoref{fig:TRA_geothermal}) \citep{emile-geay.madec_OS09}. 
    913913 
    914914% ================================================================ 
     
    955955the diffusive flux between two adjacent cells at the ocean floor is given by 
    956956\[ 
    957   % \label{eq:tra_bbl_diff} 
     957  % \label{eq:TRA_bbl_diff} 
    958958  \vect F_\sigma = A_l^\sigma \, \nabla_\sigma T 
    959959\] 
     
    963963\ie\ in the conditional form 
    964964\begin{equation} 
    965   \label{eq:tra_bbl_coef} 
     965  \label{eq:TRA_bbl_coef} 
    966966  A_l^\sigma (i,j,t) = 
    967967      \begin{cases} 
     
    973973where $A_{bbl}$ is the BBL diffusivity coefficient, given by the namelist parameter \np{rn\_ahtbbl} and 
    974974usually set to a value much larger than the one used for lateral mixing in the open ocean. 
    975 The constraint in \autoref{eq:tra_bbl_coef} implies that sigma-like diffusion only occurs when 
     975The constraint in \autoref{eq:TRA_bbl_coef} implies that sigma-like diffusion only occurs when 
    976976the density above the sea floor, at the top of the slope, is larger than in the deeper ocean 
    977 (see green arrow in \autoref{fig:bbl}). 
     977(see green arrow in \autoref{fig:TRA_bbl}). 
    978978In practice, this constraint is applied separately in the two horizontal directions, 
    979 and the density gradient in \autoref{eq:tra_bbl_coef} is evaluated with the log gradient formulation: 
     979and the density gradient in \autoref{eq:TRA_bbl_coef} is evaluated with the log gradient formulation: 
    980980\[ 
    981   % \label{eq:tra_bbl_Drho} 
     981  % \label{eq:TRA_bbl_Drho} 
    982982  \nabla_\sigma \rho / \rho = \alpha \, \nabla_\sigma T + \beta \, \nabla_\sigma S 
    983983\] 
     
    10021002    \includegraphics[width=\textwidth]{Fig_BBL_adv} 
    10031003    \caption{ 
    1004       \protect\label{fig:bbl} 
     1004      \protect\label{fig:TRA_bbl} 
    10051005      Advective/diffusive Bottom Boundary Layer. 
    10061006      The BBL parameterisation is activated when $\rho^i_{kup}$ is larger than $\rho^{i + 1}_{kdnw}$. 
     
    10261026\np{nn\_bbl\_adv}\forcode{=1}: 
    10271027the downslope velocity is chosen to be the Eulerian ocean velocity just above the topographic step 
    1028 (see black arrow in \autoref{fig:bbl}) \citep{beckmann.doscher_JPO97}. 
     1028(see black arrow in \autoref{fig:TRA_bbl}) \citep{beckmann.doscher_JPO97}. 
    10291029It is a \textit{conditional advection}, that is, advection is allowed only 
    10301030if dense water overlies less dense water on the slope (\ie\ $\nabla_\sigma \rho \cdot \nabla H < 0$) and 
     
    10361036The advection is allowed only  if dense water overlies less dense water on the slope 
    10371037(\ie\ $\nabla_\sigma \rho \cdot \nabla H < 0$). 
    1038 For example, the resulting transport of the downslope flow, here in the $i$-direction (\autoref{fig:bbl}), 
     1038For example, the resulting transport of the downslope flow, here in the $i$-direction (\autoref{fig:TRA_bbl}), 
    10391039is simply given by the following expression: 
    10401040\[ 
    1041   % \label{eq:bbl_Utr} 
     1041  % \label{eq:TRA_bbl_Utr} 
    10421042  u^{tr}_{bbl} = \gamma g \frac{\Delta \rho}{\rho_o} e_{1u} \, min ({e_{3u}}_{kup},{e_{3u}}_{kdwn}) 
    10431043\] 
     
    10531053the surrounding water at intermediate depths. 
    10541054The entrainment is replaced by the vertical mixing implicit in the advection scheme. 
    1055 Let us consider as an example the case displayed in \autoref{fig:bbl} where 
     1055Let us consider as an example the case displayed in \autoref{fig:TRA_bbl} where 
    10561056the density at level $(i,kup)$ is larger than the one at level $(i,kdwn)$. 
    10571057The advective BBL scheme modifies the tracer time tendency of the ocean cells near the topographic step by 
    1058 the downslope flow \autoref{eq:bbl_dw}, the horizontal \autoref{eq:bbl_hor} and 
    1059 the upward \autoref{eq:bbl_up} return flows as follows: 
     1058the downslope flow \autoref{eq:TRA_bbl_dw}, the horizontal \autoref{eq:TRA_bbl_hor} and 
     1059the upward \autoref{eq:TRA_bbl_up} return flows as follows: 
    10601060\begin{alignat}{3} 
    1061   \label{eq:bbl_dw} 
     1061  \label{eq:TRA_bbl_dw} 
    10621062  \partial_t T^{do}_{kdw} &\equiv \partial_t T^{do}_{kdw} 
    10631063                                &&+ \frac{u^{tr}_{bbl}}{{b_t}^{do}_{kdw}} &&\lt( T^{sh}_{kup} - T^{do}_{kdw} \rt) \\ 
    1064   \label{eq:bbl_hor} 
     1064  \label{eq:TRA_bbl_hor} 
    10651065  \partial_t T^{sh}_{kup} &\equiv \partial_t T^{sh}_{kup} 
    10661066                                &&+ \frac{u^{tr}_{bbl}}{{b_t}^{sh}_{kup}} &&\lt( T^{do}_{kup} - T^{sh}_{kup} \rt) \\ 
     
    10681068  \intertext{and for $k =kdw-1,\;..., \; kup$ :} 
    10691069  % 
    1070   \label{eq:bbl_up} 
     1070  \label{eq:TRA_bbl_up} 
    10711071  \partial_t T^{do}_{k} &\equiv \partial_t S^{do}_{k} 
    10721072                                &&+ \frac{u^{tr}_{bbl}}{{b_t}^{do}_{k}}   &&\lt( T^{do}_{k +1} - T^{sh}_{k}   \rt) 
     
    10901090In some applications it can be useful to add a Newtonian damping term into the temperature and salinity equations: 
    10911091\begin{equation} 
    1092   \label{eq:tra_dmp} 
     1092  \label{eq:TRA_dmp} 
    10931093  \begin{gathered} 
    10941094    \pd[T]{t} = \cdots - \gamma (T - T_o) \\ 
     
    11081108The DMP\_TOOLS tool is provided to allow users to generate the netcdf file. 
    11091109 
    1110 The two main cases in which \autoref{eq:tra_dmp} is used are 
     1110The two main cases in which \autoref{eq:TRA_dmp} is used are 
    11111111\textit{(a)} the specification of the boundary conditions along artificial walls of a limited domain basin and 
    11121112\textit{(b)} the computation of the velocity field associated with a given $T$-$S$ field 
     
    11461146Options are defined through the \nam{dom} namelist variables. 
    11471147The general framework for tracer time stepping is a modified leap-frog scheme \citep{leclair.madec_OM09}, 
    1148 \ie\ a three level centred time scheme associated with a Asselin time filter (cf. \autoref{sec:STP_mLF}): 
    1149 \begin{equation} 
    1150   \label{eq:tra_nxt} 
     1148\ie\ a three level centred time scheme associated with a Asselin time filter (cf. \autoref{sec:TD_mLF}): 
     1149\begin{equation} 
     1150  \label{eq:TRA_nxt} 
    11511151  \begin{alignedat}{3} 
    11521152    &(e_{3t}T)^{t + \rdt} &&= (e_{3t}T)_f^{t - \rdt} &&+ 2 \, \rdt \,e_{3t}^t \ \text{RHS}^t \\ 
     
    12631263 
    12641264  \begin{gather*} 
    1265     % \label{eq:tra_S-EOS} 
     1265    % \label{eq:TRA_S-EOS} 
    12661266    \begin{alignedat}{2} 
    12671267    &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. \\ 
     
    12721272    \text{with~} T_a = T - 10 \, ; \, S_a = S - 35 \, ; \, \rho_o = 1026~Kg/m^3 
    12731273  \end{gather*} 
    1274   where the computer name of the coefficients as well as their standard value are given in \autoref{tab:SEOS}. 
     1274  where the computer name of the coefficients as well as their standard value are given in \autoref{tab:TRA_SEOS}. 
    12751275  In fact, when choosing S-EOS, various approximation of EOS can be specified simply by 
    12761276  changing the associated coefficients. 
     
    13041304    \end{tabular} 
    13051305    \caption{ 
    1306       \protect\label{tab:SEOS} 
     1306      \protect\label{tab:TRA_SEOS} 
    13071307      Standard value of S-EOS coefficients. 
    13081308    } 
     
    13261326The expression for $N^2$  is given by: 
    13271327\[ 
    1328   % \label{eq:tra_bn2} 
     1328  % \label{eq:TRA_bn2} 
    13291329  N^2 = \frac{g}{e_{3w}} \lt( \beta \; \delta_{k + 1/2}[S] - \alpha \; \delta_{k + 1/2}[T] \rt) 
    13301330\] 
     
    13431343The freezing point of seawater is a function of salinity and pressure \citep{fofonoff.millard_bk83}: 
    13441344\begin{equation} 
    1345   \label{eq:tra_eos_fzp} 
     1345  \label{eq:TRA_eos_fzp} 
    13461346  \begin{split} 
    13471347    &T_f (S,p) = \lt( a + b \, \sqrt{S} + c \, S \rt) \, S + d \, p \\ 
     
    13511351\end{equation} 
    13521352 
    1353 \autoref{eq:tra_eos_fzp} is only used to compute the potential freezing point of sea water 
     1353\autoref{eq:TRA_eos_fzp} is only used to compute the potential freezing point of sea water 
    13541354(\ie\ referenced to the surface $p = 0$), 
    1355 thus the pressure dependent terms in \autoref{eq:tra_eos_fzp} (last term) have been dropped. 
     1355thus the pressure dependent terms in \autoref{eq:TRA_eos_fzp} (last term) have been dropped. 
    13561356The freezing point is computed through \textit{eos\_fzp}, 
    13571357a \fortran function that can be found in \mdl{eosbn2}. 
     
    13861386Before taking horizontal gradients between the tracers next to the bottom, 
    13871387a linear interpolation in the vertical is used to approximate the deeper tracer as if 
    1388 it actually lived at the depth of the shallower tracer point (\autoref{fig:Partial_step_scheme}). 
     1388it actually lived at the depth of the shallower tracer point (\autoref{fig:TRA_Partial_step_scheme}). 
    13891389For example, for temperature in the $i$-direction the needed interpolated temperature, $\widetilde T$, is: 
    13901390 
     
    13941394    \includegraphics[width=\textwidth]{Fig_partial_step_scheme} 
    13951395    \caption{ 
    1396       \protect\label{fig:Partial_step_scheme} 
     1396      \protect\label{fig:TRA_Partial_step_scheme} 
    13971397      Discretisation of the horizontal difference and average of tracers in the $z$-partial step coordinate 
    13981398      (\protect\np{ln\_zps}\forcode{=.true.}) in the case $(e3w_k^{i + 1} - e3w_k^i) > 0$. 
     
    14171417and the resulting forms for the horizontal difference and the horizontal average value of $T$ at a $U$-point are: 
    14181418\begin{equation} 
    1419   \label{eq:zps_hde} 
     1419  \label{eq:TRA_zps_hde} 
    14201420  \begin{split} 
    14211421    \delta_{i + 1/2} T       &= 
     
    14431443(in the equation of state pressure is approximated by depth, see \autoref{subsec:TRA_eos}): 
    14441444\[ 
    1445   % \label{eq:zps_hde_rho} 
     1445  % \label{eq:TRA_zps_hde_rho} 
    14461446  \widetilde \rho = \rho (\widetilde T,\widetilde S,z_u) \quad \text{where~} z_u = \min \lt( z_T^{i + 1},z_T^i \rt) 
    14471447\] 
     
    14541454Note that in almost all the advection schemes presented in this Chapter, 
    14551455both averaging and differencing operators appear. 
    1456 Yet \autoref{eq:zps_hde} has not been used in these schemes: 
     1456Yet \autoref{eq:TRA_zps_hde} has not been used in these schemes: 
    14571457in contrast to diffusion and pressure gradient computations, 
    14581458no correction for partial steps is applied for advection. 
  • NEMO/trunk/doc/latex/NEMO/subfiles/chap_ZDF.tex

    r11537 r11543  
    1818% ================================================================ 
    1919\section{Vertical mixing} 
    20 \label{sec:ZDF_zdf} 
     20\label{sec:ZDF} 
    2121 
    2222The discrete form of the ocean subgrid scale physics has been presented in 
     
    4141%(namelist parameter \np{ln\_zdfexp}\forcode{=.true.}) or a backward time stepping scheme 
    4242%(\np{ln\_zdfexp}\forcode{=.false.}) depending on the magnitude of the mixing coefficients, 
    43 %and thus of the formulation used (see \autoref{chap:STP}). 
     43%and thus of the formulation used (see \autoref{chap:TD}). 
    4444 
    4545%--------------------------------------------namzdf-------------------------------------------------------- 
     
    9292Following \citet{pacanowski.philander_JPO81}, the following formulation has been implemented: 
    9393\[ 
    94   % \label{eq:zdfric} 
     94  % \label{eq:ZDF_ric} 
    9595  \left\{ 
    9696    \begin{aligned} 
     
    151151its destruction through stratification, its vertical diffusion, and its dissipation of \citet{kolmogorov_IANS42} type: 
    152152\begin{equation} 
    153   \label{eq:zdftke_e} 
     153  \label{eq:ZDF_tke_e} 
    154154  \frac{\partial \bar{e}}{\partial t} = 
    155155  \frac{K_m}{{e_3}^2 }\;\left[ {\left( {\frac{\partial u}{\partial k}} \right)^2 
     
    161161\end{equation} 
    162162\[ 
    163   % \label{eq:zdftke_kz} 
     163  % \label{eq:ZDF_tke_kz} 
    164164  \begin{split} 
    165165    K_m &= C_k\  l_k\  \sqrt {\bar{e}\; }    \\ 
     
    175175$P_{rt}$ can be set to unity or, following \citet{blanke.delecluse_JPO93}, be a function of the local Richardson number, $R_i$: 
    176176\begin{align*} 
    177   % \label{eq:prt} 
     177  % \label{eq:ZDF_prt} 
    178178  P_{rt} = 
    179179  \begin{cases} 
     
    208208The first two are based on the following first order approximation \citep{blanke.delecluse_JPO93}: 
    209209\begin{equation} 
    210   \label{eq:tke_mxl0_1} 
     210  \label{eq:ZDF_tke_mxl0_1} 
    211211  l_k = l_\epsilon = \sqrt {2 \bar{e}\; } / N 
    212212\end{equation} 
     
    219219To overcome these drawbacks, \citet{madec.delecluse.ea_NPM98} introduces the \np{nn\_mxl}\forcode{=2, 3} cases, 
    220220which add an extra assumption concerning the vertical gradient of the computed length scale. 
    221 So, the length scales are first evaluated as in \autoref{eq:tke_mxl0_1} and then bounded such that: 
     221So, the length scales are first evaluated as in \autoref{eq:ZDF_tke_mxl0_1} and then bounded such that: 
    222222\begin{equation} 
    223   \label{eq:tke_mxl_constraint} 
     223  \label{eq:ZDF_tke_mxl_constraint} 
    224224  \frac{1}{e_3 }\left| {\frac{\partial l}{\partial k}} \right| \leq 1 
    225225  \qquad \text{with }\  l =  l_k = l_\epsilon 
    226226\end{equation} 
    227 \autoref{eq:tke_mxl_constraint} means that the vertical variations of the length scale cannot be larger than 
     227\autoref{eq:ZDF_tke_mxl_constraint} means that the vertical variations of the length scale cannot be larger than 
    228228the variations of depth. 
    229229It provides a better approximation of the \citet{gaspar.gregoris.ea_JGR90} formulation while being much less 
     
    231231In particular, it allows the length scale to be limited not only by the distance to the surface or 
    232232to the ocean bottom but also by the distance to a strongly stratified portion of the water column such as 
    233 the thermocline (\autoref{fig:mixing_length}). 
    234 In order to impose the \autoref{eq:tke_mxl_constraint} constraint, we introduce two additional length scales: 
     233the thermocline (\autoref{fig:ZDF_mixing_length}). 
     234In order to impose the \autoref{eq:ZDF_tke_mxl_constraint} constraint, we introduce two additional length scales: 
    235235$l_{up}$ and $l_{dwn}$, the upward and downward length scales, and 
    236236evaluate the dissipation and mixing length scales as 
     
    241241    \includegraphics[width=\textwidth]{Fig_mixing_length} 
    242242    \caption{ 
    243       \protect\label{fig:mixing_length} 
     243      \protect\label{fig:ZDF_mixing_length} 
    244244      Illustration of the mixing length computation. 
    245245    } 
     
    248248%>>>>>>>>>>>>>>>>>>>>>>>>>>>> 
    249249\[ 
    250   % \label{eq:tke_mxl2} 
     250  % \label{eq:ZDF_tke_mxl2} 
    251251  \begin{aligned} 
    252252    l_{up\ \ }^{(k)} &= \min \left(  l^{(k)} \ , \ l_{up}^{(k+1)} + e_{3t}^{(k)}\ \ \ \;  \right) 
     
    256256  \end{aligned} 
    257257\] 
    258 where $l^{(k)}$ is computed using \autoref{eq:tke_mxl0_1}, \ie\ $l^{(k)} = \sqrt {2 {\bar e}^{(k)} / {N^2}^{(k)} }$. 
     258where $l^{(k)}$ is computed using \autoref{eq:ZDF_tke_mxl0_1}, \ie\ $l^{(k)} = \sqrt {2 {\bar e}^{(k)} / {N^2}^{(k)} }$. 
    259259 
    260260In the \np{nn\_mxl}\forcode{=2} case, the dissipation and mixing length scales take the same value: 
     
    262262the dissipation and mixing turbulent length scales are give as in \citet{gaspar.gregoris.ea_JGR90}: 
    263263\[ 
    264   % \label{eq:tke_mxl_gaspar} 
     264  % \label{eq:ZDF_tke_mxl_gaspar} 
    265265  \begin{aligned} 
    266266    & l_k          = \sqrt{\  l_{up} \ \ l_{dwn}\ }   \\ 
     
    325325The parameterization, tuned against large-eddy simulation, includes the whole effect of LC in 
    326326an extra source term of TKE, $P_{LC}$. 
    327 The presence of $P_{LC}$ in \autoref{eq:zdftke_e}, the TKE equation, is controlled by setting \np{ln\_lc} to 
     327The presence of $P_{LC}$ in \autoref{eq:ZDF_tke_e}, the TKE equation, is controlled by setting \np{ln\_lc} to 
    328328\forcode{.true.} in the \nam{zdf\_tke} namelist. 
    329329 
     
    428428$\psi$ \citep{umlauf.burchard_JMR03, umlauf.burchard_CSR05}. 
    429429This later variable is defined as: $\psi = {C_{0\mu}}^{p} \ {\bar{e}}^{m} \ l^{n}$, 
    430 where the triplet $(p, m, n)$ value given in Tab.\autoref{tab:GLS} allows to recover a number of 
     430where the triplet $(p, m, n)$ value given in Tab.\autoref{tab:ZDF_GLS} allows to recover a number of 
    431431well-known turbulent closures ($k$-$kl$ \citep{mellor.yamada_RG82}, $k$-$\epsilon$ \citep{rodi_JGR87}, 
    432432$k$-$\omega$ \citep{wilcox_AJ88} among others \citep{umlauf.burchard_JMR03,kantha.carniel_JMR03}). 
    433433The GLS scheme is given by the following set of equations: 
    434434\begin{equation} 
    435   \label{eq:zdfgls_e} 
     435  \label{eq:ZDF_gls_e} 
    436436  \frac{\partial \bar{e}}{\partial t} = 
    437437  \frac{K_m}{\sigma_e e_3 }\;\left[ {\left( \frac{\partial u}{\partial k} \right)^2 
     
    443443 
    444444\[ 
    445   % \label{eq:zdfgls_psi} 
     445  % \label{eq:ZDF_gls_psi} 
    446446  \begin{split} 
    447447    \frac{\partial \psi}{\partial t} =& \frac{\psi}{\bar{e}} \left\{ 
     
    455455 
    456456\[ 
    457   % \label{eq:zdfgls_kz} 
     457  % \label{eq:ZDF_gls_kz} 
    458458  \begin{split} 
    459459    K_m    &= C_{\mu} \ \sqrt {\bar{e}} \ l         \\ 
     
    463463 
    464464\[ 
    465   % \label{eq:zdfgls_eps} 
     465  % \label{eq:ZDF_gls_eps} 
    466466  {\epsilon} = C_{0\mu} \,\frac{\bar {e}^{3/2}}{l} \; 
    467467\] 
     
    470470The constants $C_1$, $C_2$, $C_3$, ${\sigma_e}$, ${\sigma_{\psi}}$ and the wall function ($Fw$) depends of 
    471471the choice of the turbulence model. 
    472 Four different turbulent models are pre-defined (\autoref{tab:GLS}). 
     472Four different turbulent models are pre-defined (\autoref{tab:ZDF_GLS}). 
    473473They are made available through the \np{nn\_clo} namelist parameter. 
    474474 
     
    495495    \end{tabular} 
    496496    \caption{ 
    497       \protect\label{tab:GLS} 
     497      \protect\label{tab:ZDF_GLS} 
    498498      Set of predefined GLS parameters, or equivalently predefined turbulence models available with 
    499499      \protect\np{ln\_zdfgls}\forcode{=.true.} and controlled by the \protect\np{nn\_clos} namelist variable in \protect\nam{zdf\_gls}. 
     
    559559    \includegraphics[width=\textwidth]{Fig_ZDF_TKE_time_scheme} 
    560560    \caption{ 
    561       \protect\label{fig:TKE_time_scheme} 
     561      \protect\label{fig:ZDF_TKE_time_scheme} 
    562562      Illustration of the subgrid kinetic energy integration in GLS and TKE schemes and its links to the momentum and tracer time integration. 
    563563    } 
     
    567567 
    568568The production of turbulence by vertical shear (the first term of the right hand side of 
    569 \autoref{eq:zdftke_e}) and  \autoref{eq:zdfgls_e}) should balance the loss of kinetic energy associated with the vertical momentum diffusion 
    570 (first line in \autoref{eq:PE_zdf}). 
     569\autoref{eq:ZDF_tke_e}) and  \autoref{eq:ZDF_gls_e}) should balance the loss of kinetic energy associated with the vertical momentum diffusion 
     570(first line in \autoref{eq:MB_zdf}). 
    571571To do so a special care has to be taken for both the time and space discretization of 
    572572the kinetic energy equation \citep{burchard_OM02,marsaleix.auclair.ea_OM08}. 
    573573 
    574 Let us first address the time stepping issue. \autoref{fig:TKE_time_scheme} shows how 
     574Let us first address the time stepping issue. \autoref{fig:ZDF_TKE_time_scheme} shows how 
    575575the two-level Leap-Frog time stepping of the momentum and tracer equations interplays with 
    576576the one-level forward time stepping of the equation for $\bar{e}$. 
     
    579579summing the result vertically: 
    580580\begin{equation} 
    581   \label{eq:energ1} 
     581  \label{eq:ZDF_energ1} 
    582582  \begin{split} 
    583583    \int_{-H}^{\eta}  u^t \,\partial_z &\left( {K_m}^t \,(\partial_z u)^{t+\rdt}  \right) \,dz   \\ 
     
    587587\end{equation} 
    588588Here, the vertical diffusion of momentum is discretized backward in time with a coefficient, $K_m$, 
    589 known at time $t$ (\autoref{fig:TKE_time_scheme}), as it is required when using the TKE scheme 
    590 (see \autoref{sec:STP_forward_imp}). 
    591 The first term of the right hand side of \autoref{eq:energ1} represents the kinetic energy transfer at 
     589known at time $t$ (\autoref{fig:ZDF_TKE_time_scheme}), as it is required when using the TKE scheme 
     590(see \autoref{sec:TD_forward_imp}). 
     591The first term of the right hand side of \autoref{eq:ZDF_energ1} represents the kinetic energy transfer at 
    592592the surface (atmospheric forcing) and at the bottom (friction effect). 
    593593The second term is always negative. 
    594594It is the dissipation rate of kinetic energy, and thus minus the shear production rate of $\bar{e}$. 
    595 \autoref{eq:energ1} implies that, to be energetically consistent, 
     595\autoref{eq:ZDF_energ1} implies that, to be energetically consistent, 
    596596the production rate of $\bar{e}$ used to compute $(\bar{e})^t$ (and thus ${K_m}^t$) should be expressed as 
    597597${K_m}^{t-\rdt}\,(\partial_z u)^{t-\rdt} \,(\partial_z u)^t$ 
     
    599599 
    600600A similar consideration applies on the destruction rate of $\bar{e}$ due to stratification 
    601 (second term of the right hand side of \autoref{eq:zdftke_e} and \autoref{eq:zdfgls_e}). 
     601(second term of the right hand side of \autoref{eq:ZDF_tke_e} and \autoref{eq:ZDF_gls_e}). 
    602602This term must balance the input of potential energy resulting from vertical mixing. 
    603603The rate of change of potential energy (in 1D for the demonstration) due to vertical mixing is obtained by 
    604604multiplying the vertical density diffusion tendency by $g\,z$ and and summing the result vertically: 
    605605\begin{equation} 
    606   \label{eq:energ2} 
     606  \label{eq:ZDF_energ2} 
    607607  \begin{split} 
    608608    \int_{-H}^{\eta} g\,z\,\partial_z &\left( {K_\rho}^t \,(\partial_k \rho)^{t+\rdt}   \right) \,dz    \\ 
     
    614614\end{equation} 
    615615where we use $N^2 = -g \,\partial_k \rho / (e_3 \rho)$. 
    616 The first term of the right hand side of \autoref{eq:energ2} is always zero because 
     616The first term of the right hand side of \autoref{eq:ZDF_energ2} is always zero because 
    617617there is no diffusive flux through the ocean surface and bottom). 
    618618The second term is minus the destruction rate of  $\bar{e}$ due to stratification. 
    619 Therefore \autoref{eq:energ1} implies that, to be energetically consistent, 
    620 the product ${K_\rho}^{t-\rdt}\,(N^2)^t$ should be used in \autoref{eq:zdftke_e} and  \autoref{eq:zdfgls_e}. 
     619Therefore \autoref{eq:ZDF_energ1} implies that, to be energetically consistent, 
     620the product ${K_\rho}^{t-\rdt}\,(N^2)^t$ should be used in \autoref{eq:ZDF_tke_e} and  \autoref{eq:ZDF_gls_e}. 
    621621 
    622622Let us now address the space discretization issue. 
    623623The vertical eddy coefficients are defined at $w$-point whereas the horizontal velocity components are in 
    624 the centre of the side faces of a $t$-box in staggered C-grid (\autoref{fig:cell}). 
     624the centre of the side faces of a $t$-box in staggered C-grid (\autoref{fig:DOM_cell}). 
    625625A space averaging is thus required to obtain the shear TKE production term. 
    626 By redoing the \autoref{eq:energ1} in the 3D case, it can be shown that the product of eddy coefficient by 
     626By redoing the \autoref{eq:ZDF_energ1} in the 3D case, it can be shown that the product of eddy coefficient by 
    627627the shear at $t$ and $t-\rdt$ must be performed prior to the averaging. 
    628628Furthermore, the time variation of $e_3$ has be taken into account. 
     
    630630The above energetic considerations leads to the following final discrete form for the TKE equation: 
    631631\begin{equation} 
    632   \label{eq:zdftke_ene} 
     632  \label{eq:ZDF_tke_ene} 
    633633  \begin{split} 
    634634    \frac { (\bar{e})^t - (\bar{e})^{t-\rdt} } {\rdt}  \equiv 
     
    647647  \end{split} 
    648648\end{equation} 
    649 where the last two terms in \autoref{eq:zdftke_ene} (vertical diffusion and Kolmogorov dissipation) 
    650 are time stepped using a backward scheme (see\autoref{sec:STP_forward_imp}). 
     649where the last two terms in \autoref{eq:ZDF_tke_ene} (vertical diffusion and Kolmogorov dissipation) 
     650are time stepped using a backward scheme (see\autoref{sec:TD_forward_imp}). 
    651651Note that the Kolmogorov term has been linearized in time in order to render the implicit computation possible. 
    652652%The restart of the TKE scheme requires the storage of $\bar {e}$, $K_m$, $K_\rho$ and $l_\epsilon$ as 
    653 %they all appear in the right hand side of \autoref{eq:zdftke_ene}. 
     653%they all appear in the right hand side of \autoref{eq:ZDF_tke_ene}. 
    654654%For the latter, it is in fact the ratio $\sqrt{\bar{e}}/l_\epsilon$ which is stored. 
    655655 
     
    679679    \includegraphics[width=\textwidth]{Fig_npc} 
    680680    \caption{ 
    681       \protect\label{fig:npc} 
     681      \protect\label{fig:ZDF_npc} 
    682682      Example of an unstable density profile treated by the non penetrative convective adjustment algorithm. 
    683683      $1^{st}$ step: the initial profile is checked from the surface to the bottom. 
     
    702702(\ie\ until the mixed portion of the water column has \textit{exactly} the density of the water just below) 
    703703\citep{madec.delecluse.ea_JPO91}. 
    704 The associated algorithm is an iterative process used in the following way (\autoref{fig:npc}): 
     704The associated algorithm is an iterative process used in the following way (\autoref{fig:ZDF_npc}): 
    705705starting from the top of the ocean, the first instability is found. 
    706706Assume in the following that the instability is located between levels $k$ and $k+1$. 
     
    759759Note that the stability test is performed on both \textit{before} and \textit{now} values of $N^2$. 
    760760This removes a potential source of divergence of odd and even time step in 
    761 a leapfrog environment \citep{leclair_phd10} (see \autoref{sec:STP_mLF}). 
     761a leapfrog environment \citep{leclair_phd10} (see \autoref{sec:TD_mLF}). 
    762762 
    763763% ------------------------------------------------------------------------------------------------------------- 
     
    772772with statically unstable density profiles. 
    773773In such a case, the term corresponding to the destruction of turbulent kinetic energy through stratification in 
    774 \autoref{eq:zdftke_e} or \autoref{eq:zdfgls_e} becomes a source term, since $N^2$ is negative. 
     774\autoref{eq:ZDF_tke_e} or \autoref{eq:ZDF_gls_e} becomes a source term, since $N^2$ is negative. 
    775775It results in large values of $A_T^{vT}$ and  $A_T^{vT}$, and also of the four neighboring values at 
    776776velocity points $A_u^{vm} {and}\;A_v^{vm}$ (up to $1\;m^2s^{-1}$). 
     
    814814Diapycnal mixing of S and T are described by diapycnal diffusion coefficients 
    815815\begin{align*} 
    816   % \label{eq:zdfddm_Kz} 
     816  % \label{eq:ZDF_ddm_Kz} 
    817817  &A^{vT} = A_o^{vT}+A_f^{vT}+A_d^{vT} \\ 
    818818  &A^{vS} = A_o^{vS}+A_f^{vS}+A_d^{vS} 
     
    826826(1981): 
    827827\begin{align} 
    828   \label{eq:zdfddm_f} 
     828  \label{eq:ZDF_ddm_f} 
    829829  A_f^{vS} &= 
    830830             \begin{cases} 
     
    832832               0                              &\text{otherwise} 
    833833             \end{cases} 
    834   \\         \label{eq:zdfddm_f_T} 
     834  \\         \label{eq:ZDF_ddm_f_T} 
    835835  A_f^{vT} &= 0.7 \ A_f^{vS} / R_\rho 
    836836\end{align} 
     
    841841    \includegraphics[width=\textwidth]{Fig_zdfddm} 
    842842    \caption{ 
    843       \protect\label{fig:zdfddm} 
     843      \protect\label{fig:ZDF_ddm} 
    844844      From \citet{merryfield.holloway.ea_JPO99} : 
    845845      (a) Diapycnal diffusivities $A_f^{vT}$ and $A_f^{vS}$ for temperature and salt in regions of salt fingering. 
     
    854854%>>>>>>>>>>>>>>>>>>>>>>>>>>>> 
    855855 
    856 The factor 0.7 in \autoref{eq:zdfddm_f_T} reflects the measured ratio $\alpha F_T /\beta F_S \approx  0.7$ of 
     856The factor 0.7 in \autoref{eq:ZDF_ddm_f_T} reflects the measured ratio $\alpha F_T /\beta F_S \approx  0.7$ of 
    857857buoyancy flux of heat to buoyancy flux of salt (\eg, \citet{mcdougall.taylor_JMR84}). 
    858858Following  \citet{merryfield.holloway.ea_JPO99}, we adopt $R_c = 1.6$, $n = 6$, and $A^{\ast v} = 10^{-4}~m^2.s^{-1}$. 
     
    861861Federov (1988) is used: 
    862862\begin{align} 
    863   % \label{eq:zdfddm_d} 
     863  % \label{eq:ZDF_ddm_d} 
    864864  A_d^{vT} &= 
    865865             \begin{cases} 
     
    869869             \end{cases} 
    870870                                       \nonumber \\ 
    871   \label{eq:zdfddm_d_S} 
     871  \label{eq:ZDF_ddm_d_S} 
    872872  A_d^{vS} &= 
    873873             \begin{cases} 
     
    878878\end{align} 
    879879 
    880 The dependencies of \autoref{eq:zdfddm_f} to \autoref{eq:zdfddm_d_S} on $R_\rho$ are illustrated in 
    881 \autoref{fig:zdfddm}. 
     880The dependencies of \autoref{eq:ZDF_ddm_f} to \autoref{eq:ZDF_ddm_d_S} on $R_\rho$ are illustrated in 
     881\autoref{fig:ZDF_ddm}. 
    882882Implementing this requires computing $R_\rho$ at each grid point on every time step. 
    883883This is done in \mdl{eosbn2} at the same time as $N^2$ is computed. 
     
    891891 \label{sec:ZDF_drg} 
    892892 
    893 %--------------------------------------------nambfr-------------------------------------------------------- 
     893%--------------------------------------------namdrg-------------------------------------------------------- 
    894894% 
    895895\nlst{namdrg} 
     
    910910For the bottom boundary layer, one has: 
    911911 \[ 
    912    % \label{eq:zdfbfr_flux} 
     912   % \label{eq:ZDF_bfr_flux} 
    913913   A^{vm} \left( \partial {\textbf U}_h / \partial z \right) = {{\cal F}}_h^{\textbf U} 
    914914 \] 
     
    926926To illustrate this, consider the equation for $u$ at $k$, the last ocean level: 
    927927\begin{equation} 
    928   \label{eq:zdfdrg_flux2} 
     928  \label{eq:ZDF_drg_flux2} 
    929929  \frac{\partial u_k}{\partial t} = \frac{1}{e_{3u}} \left[ \frac{A_{uw}^{vm}}{e_{3uw}} \delta_{k+1/2}\;[u] - {\cal F}^u_h \right] \approx - \frac{{\cal F}^u_{h}}{e_{3u}} 
    930930\end{equation} 
     
    946946 These coefficients are computed in \mdl{zdfdrg} and generally take the form $c_b^{\textbf U}$ where: 
    947947\begin{equation} 
    948   \label{eq:zdfbfr_bdef} 
     948  \label{eq:ZDF_bfr_bdef} 
    949949  \frac{\partial {\textbf U_h}}{\partial t} = 
    950950  - \frac{{\cal F}^{\textbf U}_{h}}{e_{3u}} = \frac{c_b^{\textbf U}}{e_{3u}} \;{\textbf U}_h^b 
     
    963963the friction is proportional to the interior velocity (\ie\ the velocity of the first/last model level): 
    964964\[ 
    965   % \label{eq:zdfbfr_linear} 
     965  % \label{eq:ZDF_bfr_linear} 
    966966  {\cal F}_h^\textbf{U} = \frac{A^{vm}}{e_3} \; \frac{\partial \textbf{U}_h}{\partial k} = r \; \textbf{U}_h^b 
    967967\] 
     
    978978It can be changed by specifying \np{rn\_Uc0} (namelist parameter). 
    979979 
    980  For the linear friction case the drag coefficient used in the general expression \autoref{eq:zdfbfr_bdef} is: 
    981 \[ 
    982   % \label{eq:zdfbfr_linbfr_b} 
     980 For the linear friction case the drag coefficient used in the general expression \autoref{eq:ZDF_bfr_bdef} is: 
     981\[ 
     982  % \label{eq:ZDF_bfr_linbfr_b} 
    983983    c_b^T = - r 
    984984\] 
     
    10021002The non-linear bottom friction parameterisation assumes that the top/bottom friction is quadratic: 
    10031003\[ 
    1004   % \label{eq:zdfdrg_nonlinear} 
     1004  % \label{eq:ZDF_drg_nonlinear} 
    10051005  {\cal F}_h^\textbf{U} = \frac{A^{vm}}{e_3 }\frac{\partial \textbf {U}_h 
    10061006  }{\partial k}=C_D \;\sqrt {u_b ^2+v_b ^2+e_b } \;\; \textbf {U}_h^b 
     
    10181018For the non-linear friction case the term computed in \mdl{zdfdrg} is: 
    10191019\[ 
    1020   % \label{eq:zdfdrg_nonlinbfr} 
     1020  % \label{eq:ZDF_drg_nonlinbfr} 
    10211021    c_b^T = - \; C_D\;\left[ \left(\bar{u_b}^{i}\right)^2 + \left(\bar{v_b}^{j}\right)^2 + e_b \right]^{1/2} 
    10221022\] 
     
    10771077 
    10781078Since this is conditionally stable, some care needs to exercised over the choice of parameters to ensure that the implementation of explicit top/bottom friction does not induce numerical instability. 
    1079 For the purposes of stability analysis, an approximation to \autoref{eq:zdfdrg_flux2} is: 
     1079For the purposes of stability analysis, an approximation to \autoref{eq:ZDF_drg_flux2} is: 
    10801080\begin{equation} 
    1081   \label{eq:Eqn_drgstab} 
     1081  \label{eq:ZDF_Eqn_drgstab} 
    10821082  \begin{split} 
    10831083    \Delta u &= -\frac{{{\cal F}_h}^u}{e_{3u}}\;2 \rdt    \\ 
     
    10901090  |\Delta u| < \;|u| 
    10911091\] 
    1092 \noindent which, using \autoref{eq:Eqn_drgstab}, gives: 
     1092\noindent which, using \autoref{eq:ZDF_Eqn_drgstab}, gives: 
    10931093\[ 
    10941094  r\frac{2\rdt}{e_{3u}} < 1 \qquad  \Rightarrow \qquad r < \frac{e_{3u}}{2\rdt}\\ 
     
    11331133At the top (below an ice shelf cavity): 
    11341134\[ 
    1135   % \label{eq:dynzdf_drg_top} 
     1135  % \label{eq:ZDF_dynZDF__drg_top} 
    11361136  \left.{\left( {\frac{A^{vm} }{e_3 }\ \frac{\partial \textbf{U}_h}{\partial k}} \right)} \right|_{t} 
    11371137  = c_{t}^{\textbf{U}}\textbf{u}^{n+1}_{t} 
     
    11401140At the bottom (above the sea floor): 
    11411141\[ 
    1142   % \label{eq:dynzdf_drg_bot} 
     1142  % \label{eq:ZDF_dynZDF__drg_bot} 
    11431143  \left.{\left( {\frac{A^{vm} }{e_3 }\ \frac{\partial \textbf{U}_h}{\partial k}} \right)} \right|_{b} 
    11441144  = c_{b}^{\textbf{U}}\textbf{u}^{n+1}_{b} 
     
    11831183and the resulting diffusivity is obtained as 
    11841184\[ 
    1185   % \label{eq:Kwave} 
     1185  % \label{eq:ZDF_Kwave} 
    11861186  A^{vT}_{wave} =  R_f \,\frac{ \epsilon }{ \rho \, N^2 } 
    11871187\] 
     
    12421242 
    12431243\begin{equation} 
    1244   \label{eq:Bv} 
     1244  \label{eq:ZDF_Bv} 
    12451245  B_{v} = \alpha {A} {U}_{st} {exp(3kz)} 
    12461246\end{equation} 
     
    12761276criteria for a range of advection schemes. The values for the Leap-Frog with Robert 
    12771277asselin filter time-stepping (as used in NEMO) are reproduced in 
    1278 \autoref{tab:zad_Aimp_CFLcrit}. Treating the vertical advection implicitly can avoid these 
     1278\autoref{tab:ZDF_zad_Aimp_CFLcrit}. Treating the vertical advection implicitly can avoid these 
    12791279restrictions but at the cost of large dispersive errors and, possibly, large numerical 
    12801280viscosity. The adaptive-implicit vertical advection option provides a targetted use of the 
     
    12961296    \end{tabular} 
    12971297    \caption{ 
    1298       \protect\label{tab:zad_Aimp_CFLcrit} 
     1298      \protect\label{tab:ZDF_zad_Aimp_CFLcrit} 
    12991299      The advective CFL criteria for a range of spatial discretizations for the Leap-Frog with Robert Asselin filter time-stepping 
    13001300      ($\nu=0.1$) as given in \citep{lemarie.debreu.ea_OM15}. 
     
    13131313 
    13141314\begin{equation} 
    1315   \label{eq:Eqn_zad_Aimp_Courant} 
     1315  \label{eq:ZDF_Eqn_zad_Aimp_Courant} 
    13161316  \begin{split} 
    13171317    Cu &= {2 \rdt \over e^n_{3t_{ijk}}} \bigg (\big [ \texttt{Max}(w^n_{ijk},0.0) - \texttt{Min}(w^n_{ijk+1},0.0) \big ]    \\ 
     
    13261326 
    13271327\begin{align} 
    1328   \label{eq:Eqn_zad_Aimp_partition} 
     1328  \label{eq:ZDF_Eqn_zad_Aimp_partition} 
    13291329Cu_{min} &= 0.15 \nonumber \\ 
    13301330Cu_{max} &= 0.3  \nonumber \\ 
     
    13431343    \includegraphics[width=\textwidth]{Fig_ZDF_zad_Aimp_coeff} 
    13441344    \caption{ 
    1345       \protect\label{fig:zad_Aimp_coeff} 
     1345      \protect\label{fig:ZDF_zad_Aimp_coeff} 
    13461346      The value of the partitioning coefficient ($C\kern-0.14em f$) used to partition vertical velocities into parts to 
    13471347      be treated implicitly and explicitly for a range of typical Courant numbers (\forcode{ln_zad_Aimp=.true.}) 
     
    13551355 
    13561356\begin{align} 
    1357   \label{eq:Eqn_zad_Aimp_partition2} 
     1357  \label{eq:ZDF_Eqn_zad_Aimp_partition2} 
    13581358    w_{i_{ijk}} &= C\kern-0.14em f_{ijk} w_{n_{ijk}}     \nonumber \\ 
    1359     w_{n_{ijk}} &= (1-C\kern-0.14em f_{ijk}) w_{n_{ijk}}            
     1359    w_{n_{ijk}} &= (1-C\kern-0.14em f_{ijk}) w_{n_{ijk}} 
    13601360\end{align} 
    13611361 
    13621362\noindent Note that the coefficient is such that the treatment is never fully implicit; 
    1363 the three cases from \autoref{eq:Eqn_zad_Aimp_partition} can be considered as: 
     1363the three cases from \autoref{eq:ZDF_Eqn_zad_Aimp_partition} can be considered as: 
    13641364fully-explicit; mixed explicit/implicit and mostly-implicit.  With the settings shown the 
    1365 coefficient ($C\kern-0.14em f$) varies as shown in \autoref{fig:zad_Aimp_coeff}. Note with these values 
     1365coefficient ($C\kern-0.14em f$) varies as shown in \autoref{fig:ZDF_zad_Aimp_coeff}. Note with these values 
    13661366the $Cu_{cut}$ boundary between the mixed implicit-explicit treatment and 'mostly 
    13671367implicit' is 0.45 which is just below the stability limited given in 
    1368 \autoref{tab:zad_Aimp_CFLcrit}  for a 3rd order scheme. 
     1368\autoref{tab:ZDF_zad_Aimp_CFLcrit}  for a 3rd order scheme. 
    13691369 
    13701370The $w_i$ component is added to the implicit solvers for the vertical mixing in 
     
    13761376vertical fluxes are then removed since they are added by the implicit solver later on. 
    13771377 
    1378 The adaptive-implicit vertical advection option is new to NEMO at v4.0 and has yet to be  
     1378The adaptive-implicit vertical advection option is new to NEMO at v4.0 and has yet to be 
    13791379used in a wide range of simulations. The following test simulation, however, does illustrate 
    13801380the potential benefits and will hopefully encourage further testing and feedback from users: 
     
    13841384    \includegraphics[width=\textwidth]{Fig_ZDF_zad_Aimp_overflow_frames} 
    13851385    \caption{ 
    1386       \protect\label{fig:zad_Aimp_overflow_frames} 
     1386      \protect\label{fig:ZDF_zad_Aimp_overflow_frames} 
    13871387      A time-series of temperature vertical cross-sections for the OVERFLOW test case. These results are for the default 
    13881388      settings with \forcode{nn_rdt=10.0} and without adaptive implicit vertical advection (\forcode{ln_zad_Aimp=.false.}). 
     
    14081408\noindent which were chosen to provide a slightly more stable and less noisy solution. The 
    14091409result when using the default value of \forcode{nn_rdt=10.} without adaptive-implicit 
    1410 vertical velocity is illustrated in \autoref{fig:zad_Aimp_overflow_frames}. The mass of 
     1410vertical velocity is illustrated in \autoref{fig:ZDF_zad_Aimp_overflow_frames}. The mass of 
    14111411cold water, initially sitting on the shelf, moves down the slope and forms a 
    14121412bottom-trapped, dense plume. Even with these extra physics choices the model is close to 
     
    14181418 
    14191419The results with \forcode{ln_zad_Aimp=.true.} and a variety of model timesteps 
    1420 are shown in \autoref{fig:zad_Aimp_overflow_all_rdt} (together with the equivalent 
     1420are shown in \autoref{fig:ZDF_zad_Aimp_overflow_all_rdt} (together with the equivalent 
    14211421frames from the base run).  In this simple example the use of the adaptive-implicit 
    14221422vertcal advection scheme has enabled a 12x increase in the model timestep without 
     
    14341434\autoref{sec:MISC_opt} for activation details). 
    14351435 
    1436 \autoref{fig:zad_Aimp_maxCf} shows examples of the maximum partitioning coefficient for 
     1436\autoref{fig:ZDF_zad_Aimp_maxCf} shows examples of the maximum partitioning coefficient for 
    14371437the various overflow tests.  Note that the adaptive-implicit vertical advection scheme is 
    14381438active even in the base run with \forcode{nn_rdt=10.0s} adding to the evidence that the 
     
    14411441oscillatory nature of this measure appears to be linked to the progress of the plume front 
    14421442as each cusp is associated with the location of the maximum shifting to the adjacent cell. 
    1443 This is illustrated in \autoref{fig:zad_Aimp_maxCf_loc} where the i- and k- locations of the 
     1443This is illustrated in \autoref{fig:ZDF_zad_Aimp_maxCf_loc} where the i- and k- locations of the 
    14441444maximum have been overlaid for the base run case. 
    14451445 
     
    14631463    \includegraphics[width=\textwidth]{Fig_ZDF_zad_Aimp_overflow_all_rdt} 
    14641464    \caption{ 
    1465       \protect\label{fig:zad_Aimp_overflow_all_rdt} 
    1466       Sample temperature vertical cross-sections from mid- and end-run using different values for \forcode{nn_rdt}  
     1465      \protect\label{fig:ZDF_zad_Aimp_overflow_all_rdt} 
     1466      Sample temperature vertical cross-sections from mid- and end-run using different values for \forcode{nn_rdt} 
    14671467      and with or without adaptive implicit vertical advection. Without the adaptive implicit vertical advection only 
    14681468      the run with the shortest timestep is able to run to completion. Note also that the colour-scale has been 
     
    14761476    \includegraphics[width=\textwidth]{Fig_ZDF_zad_Aimp_maxCf} 
    14771477    \caption{ 
    1478       \protect\label{fig:zad_Aimp_maxCf} 
     1478      \protect\label{fig:ZDF_zad_Aimp_maxCf} 
    14791479      The maximum partitioning coefficient during a series of test runs with increasing model timestep length. 
    1480       At the larger timesteps, the vertical velocity is treated mostly implicitly at some location throughout  
     1480      At the larger timesteps, the vertical velocity is treated mostly implicitly at some location throughout 
    14811481      the run. 
    14821482    } 
     
    14881488    \includegraphics[width=\textwidth]{Fig_ZDF_zad_Aimp_maxCf_loc} 
    14891489    \caption{ 
    1490       \protect\label{fig:zad_Aimp_maxCf_loc} 
     1490      \protect\label{fig:ZDF_zad_Aimp_maxCf_loc} 
    14911491      The maximum partitioning coefficient for the  \forcode{nn_rdt=10.0s} case overlaid with  information on the gridcell i- and k- 
    1492       locations of the maximum value.  
     1492      locations of the maximum value. 
    14931493    } 
    14941494  \end{center} 
  • NEMO/trunk/doc/latex/NEMO/subfiles/chap_cfgs.tex

    r11541 r11543  
    66% ================================================================ 
    77\chapter{Configurations} 
    8 \label{chap:CFG} 
     8\label{chap:CFGS} 
    99 
    1010\chaptertoc 
     
    1616% ================================================================ 
    1717\section{Introduction} 
    18 \label{sec:CFG_intro} 
     18\label{sec:CFGS_intro} 
    1919 
    2020The purpose of this part of the manual is to introduce the \NEMO\ reference configurations. 
     
    3636\section[C1D: 1D Water column model (\texttt{\textbf{key\_c1d}})] 
    3737{C1D: 1D Water column model (\protect\key{c1d})} 
    38 \label{sec:CFG_c1d} 
     38\label{sec:CFGS_c1d} 
    3939 
    4040The 1D model option simulates a stand alone water column within the 3D \NEMO\ system. 
     
    7676% ================================================================ 
    7777\section{ORCA family: global ocean with tripolar grid} 
    78 \label{sec:CFG_orca} 
     78\label{sec:CFGS_orca} 
    7979 
    8080The ORCA family is a series of global ocean configurations that are run together with 
     
    9292    \includegraphics[width=\textwidth]{Fig_ORCA_NH_mesh} 
    9393    \caption{ 
    94       \protect\label{fig:MISC_ORCA_msh} 
     94      \protect\label{fig:CFGS_ORCA_msh} 
    9595      ORCA mesh conception. 
    9696      The departure from an isotropic Mercator grid start poleward of 20\deg{N}. 
     
    108108% ------------------------------------------------------------------------------------------------------------- 
    109109\subsection{ORCA tripolar grid} 
    110 \label{subsec:CFG_orca_grid} 
     110\label{subsec:CFGS_orca_grid} 
    111111 
    112112The ORCA grid is a tripolar grid based on the semi-analytical method of \citet{madec.imbard_CD96}. 
     
    116116computing the associated set of mesh meridians, and projecting the resulting mesh onto the sphere. 
    117117The set of mesh parallels used is a series of embedded ellipses which foci are the two mesh north poles 
    118 (\autoref{fig:MISC_ORCA_msh}). 
     118(\autoref{fig:CFGS_ORCA_msh}). 
    119119The resulting mesh presents no loss of continuity in either the mesh lines or the scale factors, 
    120120or even the scale factor derivatives over the whole ocean domain, as the mesh is not a composite mesh. 
     
    125125    \includegraphics[width=\textwidth]{Fig_ORCA_aniso} 
    126126    \caption { 
    127       \protect\label{fig:MISC_ORCA_e1e2} 
     127      \protect\label{fig:CFGS_ORCA_e1e2} 
    128128      \textit{Top}: Horizontal scale factors ($e_1$, $e_2$) and 
    129129      \textit{Bottom}: ratio of anisotropy ($e_1 / e_2$) 
     
    143143(especially in area of strong eddy activities such as the Gulf Stream) and keeping the smallest scale factor in 
    144144the northern hemisphere larger than the smallest one in the southern hemisphere. 
    145 The resulting mesh is shown in \autoref{fig:MISC_ORCA_msh} and \autoref{fig:MISC_ORCA_e1e2} for 
     145The resulting mesh is shown in \autoref{fig:CFGS_ORCA_msh} and \autoref{fig:CFGS_ORCA_e1e2} for 
    146146a half a degree grid (ORCA\_R05). 
    147147The smallest ocean scale factor is found in along Antarctica, 
     
    152152% ------------------------------------------------------------------------------------------------------------- 
    153153\subsection{ORCA pre-defined resolution} 
    154 \label{subsec:CFG_orca_resolution} 
     154\label{subsec:CFGS_orca_resolution} 
    155155 
    156156The \NEMO\ system is provided with five built-in ORCA configurations which differ in the horizontal resolution. 
     
    159159which sets the grid size and configuration name parameters. 
    160160The \NEMO\ System Team provides only ORCA2 domain input file "\ifile{ORCA\_R2\_zps\_domcfg}" file 
    161 (\autoref{tab:ORCA}). 
     161(\autoref{tab:CFGS_ORCA}). 
    162162 
    163163%--------------------------------------------------TABLE-------------------------------------------------- 
     
    175175    \end{tabular} 
    176176    \caption{ 
    177       \protect\label{tab:ORCA} 
     177      \protect\label{tab:CFGS_ORCA} 
    178178      Domain size of ORCA family configurations. 
    179179      The flag for configurations of ORCA family need to be set in \textit{domain\_cfg} file. 
     
    184184 
    185185 
    186 The ORCA\_R2 configuration has the following specificity: starting from a 2\deg~ORCA mesh, 
     186The ORCA\_R2 configuration has the following specificity: starting from a 2\deg\ ORCA mesh, 
    187187local mesh refinements were applied to the Mediterranean, Red, Black and Caspian Seas, 
    188 so that the resolution is 1\deg~ there. 
     188so that the resolution is 1\deg\ there. 
    189189A local transformation were also applied with in the Tropics in order to refine the meridional resolution up to 
    190 0.5\deg~ at the Equator. 
     1900.5\deg\ at the Equator. 
    191191 
    192192The ORCA\_R1 configuration has only a local tropical transformation to refine the meridional resolution up to 
    193 1/3\deg~at the Equator. 
     1931/3\deg\ at the Equator. 
    194194Note that the tropical mesh refinements in ORCA\_R2 and R1 strongly increases the mesh anisotropy there. 
    195195 
     
    198198For ORCA\_R1 and R025, setting the configuration key to 75 allows to use 75 vertical levels, otherwise 46 are used. 
    199199In the other ORCA configurations, 31 levels are used 
    200 (see \autoref{tab:orca_zgr}). %\sfcomment{HERE I need to put new table for ORCA2 values} and \autoref{fig:zgr}). 
     200(see \autoref{tab:CFGS_ORCA}). %\sfcomment{HERE I need to put new table for ORCA2 values} and \autoref{fig:DOM_zgr_e3}). 
    201201 
    202202Only the ORCA\_R2 is provided with all its input files in the \NEMO\ distribution. 
     
    207207 
    208208This version of ORCA\_R2 has 31 levels in the vertical, with the highest resolution (10m) in the upper 150m 
    209 (see \autoref{tab:orca_zgr} and \autoref{fig:zgr}). 
     209(see \autoref{tab:CFGS_ORCA} and \autoref{fig:DOM_zgr_e3}). 
    210210The bottom topography and the coastlines are derived from the global atlas of Smith and Sandwell (1997). 
    211 The default forcing uses the boundary forcing from \citet{large.yeager_rpt04} (see \autoref{subsec:SBC_blk_core}), 
     211The default forcing uses the boundary forcing from \citet{large.yeager_rpt04} (see \autoref{subsec:SBC_blk_ocean}), 
    212212which was developed for the purpose of running global coupled ocean-ice simulations without 
    213213an interactive atmosphere. 
     
    226226% ------------------------------------------------------------------------------------------------------------- 
    227227\section{GYRE family: double gyre basin} 
    228 \label{sec:CFG_gyre} 
     228\label{sec:CFGS_gyre} 
    229229 
    230230The GYRE configuration \citep{levy.klein.ea_OM10} has been built to 
     
    254254 
    255255The GYRE configuration is set like an analytical configuration. 
    256 Through \np{ln\_read\_cfg}\forcode{=.false.} in \nam{cfg} namelist defined in 
     256Through \np{ln\_read\_cfg}\forcode{ = .false.} in \nam{cfg} namelist defined in 
    257257the reference configuration \path{./cfgs/GYRE_PISCES/EXPREF/namelist_cfg} 
    258258analytical definition of grid in GYRE is done in usrdef\_hrg, usrdef\_zgr routines. 
     
    266266Obviously, the namelist parameters have to be adjusted to the chosen resolution, 
    267267see the Configurations pages on the \NEMO\ web site (\NEMO\ Configurations). 
    268 In the vertical, GYRE uses the default 30 ocean levels (\jp{jpk}\forcode{=31}) (\autoref{fig:zgr}). 
     268In the vertical, GYRE uses the default 30 ocean levels (\jp{jpk}\forcode{ = 31}) (\autoref{fig:DOM_zgr_e3}). 
    269269 
    270270The GYRE configuration is also used in benchmark test as it is very simple to increase its resolution and 
     
    272272For example, keeping a same model size on each processor while increasing the number of processor used is very easy, 
    273273even though the physical integrity of the solution can be compromised. 
    274 Benchmark is activate via \np{ln\_bench}\forcode{=.true.} in \nam{usr\_def} in 
     274Benchmark is activate via \np{ln\_bench}\forcode{ = .true.} in \nam{usr\_def} in 
    275275namelist \path{./cfgs/GYRE_PISCES/EXPREF/namelist_cfg}. 
    276276 
     
    280280    \includegraphics[width=\textwidth]{Fig_GYRE} 
    281281    \caption{ 
    282       \protect\label{fig:GYRE} 
     282      \protect\label{fig:CFGS_GYRE} 
    283283      Snapshot of relative vorticity at the surface of the model domain in GYRE R9, R27 and R54. 
    284284      From \citet{levy.klein.ea_OM10}. 
     
    292292% ------------------------------------------------------------------------------------------------------------- 
    293293\section{AMM: atlantic margin configuration} 
    294 \label{sec:MISC_config_AMM} 
     294\label{sec:CFGS_config_AMM} 
    295295 
    296296The AMM, Atlantic Margins Model, is a regional model covering the Northwest European Shelf domain on 
  • NEMO/trunk/doc/latex/NEMO/subfiles/chap_misc.tex

    r11435 r11543  
    110110    \includegraphics[width=\textwidth]{Fig_closea_mask_example} 
    111111    \caption{ 
    112       \protect\label{fig:closea_mask_example} 
     112      \protect\label{fig:MISC_closea_mask_example} 
    113113      Example of mask fields for the closea module. \textit{Left}: a 
    114114      closea\_mask field; \textit{Right}: a closea\_mask\_rnf 
     
    159159configuration file and ln\_closea=.true. in namelist namcfg.} Each 
    160160inland sea or group of inland seas is set to a positive integer value 
    161 in the closea\_mask field (see Figure \ref{fig:closea_mask_example} 
     161in the closea\_mask field (see \autoref{fig:MISC_closea_mask_example} 
    162162for an example). The net surface flux over each inland sea or group of 
    163163inland seas is set to zero each timestep and the residual flux is 
     
    174174by the closea\_mask\_rnf field. Each mapping is defined by a positive 
    175175integer value for the inland sea(s) and the corresponding runoff 
    176 points. An example is given in Figure 
    177 \ref{fig:closea_mask_example}. If no mapping is provided for a 
     176points. An example is given in 
     177\autoref{fig:MISC_closea_mask_example}. If no mapping is provided for a 
    178178particular inland sea then the residual is spread over the global 
    179179ocean.} 
  • NEMO/trunk/doc/latex/NEMO/subfiles/chap_model_basics.tex

    r11435 r11543  
    88% ================================================================ 
    99\chapter{Model Basics} 
    10 \label{chap:PE} 
     10\label{chap:MB} 
    1111 
    1212\chaptertoc 
     
    1818% ================================================================ 
    1919\section{Primitive equations} 
    20 \label{sec:PE_PE} 
     20\label{sec:MB_PE} 
    2121 
    2222% ------------------------------------------------------------------------------------------------------------- 
     
    2525 
    2626\subsection{Vector invariant formulation} 
    27 \label{subsec:PE_Vector} 
     27\label{subsec:MB_PE_vector} 
    2828 
    2929The ocean is a fluid that can be described to a good approximation by the primitive equations, 
     
    4848  the buoyancy force 
    4949  \begin{equation} 
    50     \label{eq:PE_eos} 
     50    \label{eq:MB_PE_eos} 
    5151    \rho = \rho \ (T,S,p) 
    5252  \end{equation} 
     
    5757  convective processes must be parameterized instead) 
    5858  \begin{equation} 
    59     \label{eq:PE_hydrostatic} 
     59    \label{eq:MB_PE_hydrostatic} 
    6060    \pd[p]{z} = - \rho \ g 
    6161  \end{equation} 
     
    6464  is assumed to be zero. 
    6565  \begin{equation} 
    66     \label{eq:PE_continuity} 
     66    \label{eq:MB_PE_continuity} 
    6767    \nabla \cdot \vect U = 0 
    6868  \end{equation} 
     
    8585the following equations: 
    8686\begin{subequations} 
    87   \label{eq:PE} 
     87  \label{eq:MB_PE} 
    8888  \begin{gather} 
    8989    \intertext{$-$ the momentum balance} 
    90     \label{eq:PE_dyn} 
     90    \label{eq:MB_PE_dyn} 
    9191    \pd[\vect U_h]{t} = - \lt[ (\nabla \times \vect U) \times \vect U + \frac{1}{2} \nabla \lt( \vect U^2 \rt) \rt]_h 
    9292                        - f \; k \times \vect U_h - \frac{1}{\rho_o} \nabla_h p 
    9393                        + \vect D^{\vect U} + \vect F^{\vect U} \\ 
    9494    \intertext{$-$ the heat and salt conservation equations} 
    95     \label{eq:PE_tra_T} 
     95    \label{eq:MB_PE_tra_T} 
    9696    \pd[T]{t} = - \nabla \cdot (T \ \vect U) + D^T + F^T \\ 
    97     \label{eq:PE_tra_S} 
     97    \label{eq:MB_PE_tra_S} 
    9898    \pd[S]{t} = - \nabla \cdot (S \ \vect U) + D^S + F^S 
    9999  \end{gather} 
     
    101101where $\nabla$ is the generalised derivative vector operator in $(i,j,k)$ directions, $t$ is the time, 
    102102$z$ is the vertical coordinate, $\rho$ is the \textit{in situ} density given by the equation of state 
    103 (\autoref{eq:PE_eos}), $\rho_o$ is a reference density, $p$ the pressure, 
     103(\autoref{eq:MB_PE_eos}), $\rho_o$ is a reference density, $p$ the pressure, 
    104104$f = 2 \vect \Omega \cdot k$ is the Coriolis acceleration 
    105105(where $\vect \Omega$ is the Earth's angular velocity vector), and $g$ is the gravitational acceleration. 
    106106$\vect D^{\vect U}$, $D^T$ and $D^S$ are the parameterisations of small-scale physics for momentum, 
    107107temperature and salinity, and $\vect F^{\vect U}$, $F^T$ and $F^S$ surface forcing terms. 
    108 Their nature and formulation are discussed in \autoref{sec:PE_zdf_ldf} and \autoref{subsec:PE_boundary_condition}. 
     108Their nature and formulation are discussed in \autoref{sec:MB_zdf_ldf} and \autoref{subsec:MB_boundary_condition}. 
    109109 
    110110% ------------------------------------------------------------------------------------------------------------- 
     
    112112% ------------------------------------------------------------------------------------------------------------- 
    113113\subsection{Boundary conditions} 
    114 \label{subsec:PE_boundary_condition} 
     114\label{subsec:MB_boundary_condition} 
    115115 
    116116An ocean is bounded by complex coastlines, bottom topography at its base and 
     
    120120(discretisation can introduce additional artificial ``side-wall'' boundaries). 
    121121Both $H$ and $\eta$ are referenced to a surface of constant geopotential (\ie\ a mean sea surface height) on which $z = 0$. 
    122 (\autoref{fig:ocean_bc}). 
     122(\autoref{fig:MB_ocean_bc}). 
    123123Through these two boundaries, the ocean can exchange fluxes of heat, fresh water, salt, and momentum with 
    124124the solid earth, the continental margins, the sea ice and the atmosphere. 
     
    133133    \includegraphics[width=\textwidth]{Fig_I_ocean_bc} 
    134134    \caption{ 
    135       \protect\label{fig:ocean_bc} 
     135      \protect\label{fig:MB_ocean_bc} 
    136136      The ocean is bounded by two surfaces, $z = - H(i,j)$ and $z = \eta(i,j,t)$, 
    137137      where $H$ is the depth of the sea floor and $\eta$ the height of the sea surface. 
     
    164164  can be expressed as: 
    165165  \begin{equation} 
    166     \label{eq:PE_w_bbc} 
     166    \label{eq:MB_w_bbc} 
    167167    w = - \vect U_h \cdot \nabla_h (H) 
    168168  \end{equation} 
     
    171171  It must be parameterized in terms of turbulent fluxes using bottom and/or lateral boundary conditions. 
    172172  Its specification depends on the nature of the physical parameterisation used for 
    173   $\vect D^{\vect U}$ in \autoref{eq:PE_dyn}. 
    174   It is discussed in \autoref{eq:PE_zdf}.% and Chap. III.6 to 9. 
     173  $\vect D^{\vect U}$ in \autoref{eq:MB_PE_dyn}. 
     174  It is discussed in \autoref{eq:MB_zdf}.% and Chap. III.6 to 9. 
    175175\item[Atmosphere - ocean interface:] 
    176176  the kinematic surface condition plus the mass flux of fresh water PE (the precipitation minus evaporation budget) 
    177177  leads to: 
    178178  \[ 
    179     % \label{eq:PE_w_sbc} 
     179    % \label{eq:MB_w_sbc} 
    180180    w = \pd[\eta]{t} + \lt. \vect U_h \rt|_{z = \eta} \cdot \nabla_h (\eta) + P - E 
    181181  \] 
     
    194194% ================================================================ 
    195195\section{Horizontal pressure gradient} 
    196 \label{sec:PE_hor_pg} 
     196\label{sec:MB_hor_pg} 
    197197 
    198198% ------------------------------------------------------------------------------------------------------------- 
     
    200200% ------------------------------------------------------------------------------------------------------------- 
    201201\subsection{Pressure formulation} 
    202 \label{subsec:PE_p_formulation} 
     202\label{subsec:MB_p_formulation} 
    203203 
    204204The total pressure at a given depth $z$ is composed of a surface pressure $p_s$ at 
    205205a reference geopotential surface ($z = 0$) and a hydrostatic pressure $p_h$ such that: 
    206206$p(i,j,k,t) = p_s(i,j,t) + p_h(i,j,k,t)$. 
    207 The latter is computed by integrating (\autoref{eq:PE_hydrostatic}), 
    208 assuming that pressure in decibars can be approximated by depth in meters in (\autoref{eq:PE_eos}). 
     207The latter is computed by integrating (\autoref{eq:MB_PE_hydrostatic}), 
     208assuming that pressure in decibars can be approximated by depth in meters in (\autoref{eq:MB_PE_eos}). 
    209209The hydrostatic pressure is then given by: 
    210210\[ 
    211   % \label{eq:PE_pressure} 
     211  % \label{eq:MB_pressure} 
    212212  p_h (i,j,z,t) = \int_{\varsigma = z}^{\varsigma = 0} g \; \rho (T,S,\varsigma) \; d \varsigma 
    213213\] 
     
    234234% ------------------------------------------------------------------------------------------------------------- 
    235235\subsection{Free surface formulation} 
    236 \label{subsec:PE_free_surface} 
     236\label{subsec:MB_free_surface} 
    237237 
    238238In the free surface formulation, a variable $\eta$, the sea-surface height, 
    239239is introduced which describes the shape of the air-sea interface. 
    240240This variable is solution of a prognostic equation which is established by forming the vertical average of 
    241 the kinematic surface condition (\autoref{eq:PE_w_bbc}): 
     241the kinematic surface condition (\autoref{eq:MB_w_bbc}): 
    242242\begin{equation} 
    243   \label{eq:PE_ssh} 
     243  \label{eq:MB_ssh} 
    244244  \pd[\eta]{t} = - D + P - E \quad \text{where} \quad D = \nabla \cdot \lt[ (H + \eta) \; \overline{U}_h \, \rt] 
    245245\end{equation} 
    246 and using (\autoref{eq:PE_hydrostatic}) the surface pressure is given by: $p_s = \rho \, g \, \eta$. 
     246and using (\autoref{eq:MB_PE_hydrostatic}) the surface pressure is given by: $p_s = \rho \, g \, \eta$. 
    247247 
    248248Allowing the air-sea interface to move introduces the external gravity waves (EGWs) as 
     
    257257the baroclinic structure of the ocean (internal waves) possibly in shallow seas, 
    258258then a non linear free surface is the most appropriate. 
    259 This means that no approximation is made in \autoref{eq:PE_ssh} and that 
     259This means that no approximation is made in \autoref{eq:MB_ssh} and that 
    260260the variation of the ocean volume is fully taken into account. 
    261261Note that in order to study the fast time scales associated with EGWs it is necessary to 
     
    268268not altering the slow barotropic Rossby waves. 
    269269If further, an approximative conservation of heat and salt contents is sufficient for the problem solved, 
    270 then it is sufficient to solve a linearized version of \autoref{eq:PE_ssh}, 
     270then it is sufficient to solve a linearized version of \autoref{eq:MB_ssh}, 
    271271which still allows to take into account freshwater fluxes applied at the ocean surface \citep{roullet.madec_JGR00}. 
    272272Nevertheless, with the linearization, an exact conservation of heat and salt contents is lost. 
     
    286286% ================================================================ 
    287287\section{Curvilinear \textit{z-}coordinate system} 
    288 \label{sec:PE_zco} 
     288\label{sec:MB_zco} 
    289289 
    290290% ------------------------------------------------------------------------------------------------------------- 
     
    292292% ------------------------------------------------------------------------------------------------------------- 
    293293\subsection{Tensorial formalism} 
    294 \label{subsec:PE_tensorial} 
     294\label{subsec:MB_tensorial} 
    295295 
    296296In many ocean circulation problems, the flow field has regions of enhanced dynamics 
     
    315315$(i,j,k)$ linked to the earth such that 
    316316$k$ is the local upward vector and $(i,j)$ are two vectors orthogonal to $k$, 
    317 \ie\ along geopotential surfaces (\autoref{fig:referential}). 
     317\ie\ along geopotential surfaces (\autoref{fig:MB_referential}). 
    318318Let $(\lambda,\varphi,z)$ be the geographical coordinate system in which a position is defined by 
    319319the latitude $\varphi(i,j)$, the longitude $\lambda(i,j)$ and 
    320320the distance from the centre of the earth $a + z(k)$ where $a$ is the earth's radius and 
    321 $z$ the altitude above a reference sea level (\autoref{fig:referential}). 
     321$z$ the altitude above a reference sea level (\autoref{fig:MB_referential}). 
    322322The local deformation of the curvilinear coordinate system is given by $e_1$, $e_2$ and $e_3$, 
    323323the three scale factors: 
    324324\begin{equation} 
    325   \label{eq:scale_factors} 
     325  \label{eq:MB_scale_factors} 
    326326  \begin{aligned} 
    327327    e_1 &= (a + z) \lt[ \lt( \pd[\lambda]{i} \cos \varphi \rt)^2 + \lt( \pd[\varphi]{i} \rt)^2 \rt]^{1/2} \\ 
     
    336336    \includegraphics[width=\textwidth]{Fig_I_earth_referential} 
    337337    \caption{ 
    338       \protect\label{fig:referential} 
     338      \protect\label{fig:MB_referential} 
    339339      the geographical coordinate system $(\lambda,\varphi,z)$ and the curvilinear 
    340340      coordinate system $(i,j,k)$. 
     
    345345 
    346346Since the ocean depth is far smaller than the earth's radius, $a + z$, can be replaced by $a$ in 
    347 (\autoref{eq:scale_factors}) (thin-shell approximation). 
     347(\autoref{eq:MB_scale_factors}) (thin-shell approximation). 
    348348The resulting horizontal scale factors $e_1$, $e_2$  are independent of $k$ while 
    349349the vertical scale factor is a single function of $k$ as $k$ is parallel to $z$. 
    350350The scalar and vector operators that appear in the primitive equations 
    351 (\autoref{eq:PE_dyn} to \autoref{eq:PE_eos}) can then be written in the tensorial form, 
     351(\autoref{eq:MB_PE_dyn} to \autoref{eq:MB_PE_eos}) can then be written in the tensorial form, 
    352352invariant in any orthogonal horizontal curvilinear coordinate system transformation: 
    353353\begin{subequations} 
    354   % \label{eq:PE_discrete_operators} 
     354  % \label{eq:MB_discrete_operators} 
    355355  \begin{gather} 
    356     \label{eq:PE_grad} 
     356    \label{eq:MB_grad} 
    357357    \nabla q =   \frac{1}{e_1} \pd[q]{i} \; \vect i 
    358358               + \frac{1}{e_2} \pd[q]{j} \; \vect j 
    359359               + \frac{1}{e_3} \pd[q]{k} \; \vect k \\ 
    360     \label{eq:PE_div} 
     360    \label{eq:MB_div} 
    361361    \nabla \cdot \vect A =   \frac{1}{e_1 \; e_2} \lt[ \pd[(e_2 \; a_1)]{\partial i} + \pd[(e_1 \; a_2)]{j} \rt] 
    362362                           + \frac{1}{e_3} \lt[ \pd[a_3]{k} \rt] 
    363363  \end{gather} 
    364364  \begin{multline} 
    365     \label{eq:PE_curl} 
     365    \label{eq:MB_curl} 
    366366      \nabla \times \vect{A} =   \lt[ \frac{1}{e_2} \pd[a_3]{j} - \frac{1}{e_3} \pd[a_2]{k}   \rt] \vect i \\ 
    367367                               + \lt[ \frac{1}{e_3} \pd[a_1]{k} - \frac{1}{e_1} \pd[a_3]{i}   \rt] \vect j \\ 
     
    369369  \end{multline} 
    370370  \begin{gather} 
    371     \label{eq:PE_lap} 
     371    \label{eq:MB_lap} 
    372372    \Delta q = \nabla \cdot (\nabla q) \\ 
    373     \label{eq:PE_lap_vector} 
     373    \label{eq:MB_lap_vector} 
    374374    \Delta \vect A = \nabla (\nabla \cdot \vect A) - \nabla \times (\nabla \times \vect A) 
    375375  \end{gather} 
     
    381381% ------------------------------------------------------------------------------------------------------------- 
    382382\subsection{Continuous model equations} 
    383 \label{subsec:PE_zco_Eq} 
     383\label{subsec:MB_zco_Eq} 
    384384 
    385385In order to express the Primitive Equations in tensorial formalism, 
    386386it is necessary to compute the horizontal component of the non-linear and viscous terms of the equation using 
    387 \autoref{eq:PE_grad}) to \autoref{eq:PE_lap_vector}. 
     387\autoref{eq:MB_grad}) to \autoref{eq:MB_lap_vector}. 
    388388Let us set $\vect U = (u,v,w) = \vect U_h + w \; \vect k $, the velocity in the $(i,j,k)$ coordinates system and 
    389389define the relative vorticity $\zeta$ and the divergence of the horizontal velocity field $\chi$, by: 
    390390\begin{gather} 
    391   \label{eq:PE_curl_Uh} 
     391  \label{eq:MB_curl_Uh} 
    392392  \zeta = \frac{1}{e_1 e_2} \lt[ \pd[(e_2 \, v)]{i} - \pd[(e_1 \, u)]{j} \rt] \\ 
    393   \label{eq:PE_div_Uh} 
     393  \label{eq:MB_div_Uh} 
    394394  \chi  = \frac{1}{e_1 e_2} \lt[ \pd[(e_2 \, u)]{i} + \pd[(e_1 \, v)]{j} \rt] 
    395395\end{gather} 
     
    397397Using again the fact that the horizontal scale factors $e_1$ and $e_2$ are independent of $k$ and that 
    398398$e_3$  is a function of the single variable $k$, 
    399 $NLT$ the nonlinear term of \autoref{eq:PE_dyn} can be transformed as follows: 
     399$NLT$ the nonlinear term of \autoref{eq:MB_PE_dyn} can be transformed as follows: 
    400400\begin{alignat*}{2} 
    401401  &NLT &=   &\lt[ (\nabla \times {\vect U}) \times {\vect U} + \frac{1}{2} \nabla \lt( {\vect U}^2 \rt) \rt]_h \\ 
     
    438438\end{alignat*} 
    439439The last term of the right hand side is obviously zero, and thus the nonlinear term of 
    440 \autoref{eq:PE_dyn} is written in the $(i,j,k)$ coordinate system: 
     440\autoref{eq:MB_PE_dyn} is written in the $(i,j,k)$ coordinate system: 
    441441\begin{equation} 
    442   \label{eq:PE_vector_form} 
     442  \label{eq:MB_vector_form} 
    443443  NLT =   \zeta \; \vect k \times \vect U_h + \frac{1}{2} \nabla_h \lt( \vect U_h^2 \rt) 
    444444        + \frac{1}{e_3} w \pd[\vect U_h]{k} 
     
    448448For some purposes, it can be advantageous to write this term in the so-called flux form, 
    449449\ie\ to write it as the divergence of fluxes. 
    450 For example, the first component of \autoref{eq:PE_vector_form} (the $i$-component) is transformed as follows: 
     450For example, the first component of \autoref{eq:MB_vector_form} (the $i$-component) is transformed as follows: 
    451451\begin{alignat*}{2} 
    452452  &NLT_i &= &- \zeta \; v + \frac{1}{2 \; e_1} \pd[ (u^2 + v^2) ]{i} + \frac{1}{e_3} w \ \pd[u]{k} \\ 
     
    473473The flux form of the momentum advection term is therefore given by: 
    474474\begin{equation} 
    475   \label{eq:PE_flux_form} 
     475  \label{eq:MB_flux_form} 
    476476  NLT =   \nabla \cdot \lt( 
    477477    \begin{array}{*{20}c} 
     
    488488The latter is called the \textit{metric} term and can be viewed as a modification of the Coriolis parameter: 
    489489\[ 
    490   % \label{eq:PE_cor+metric} 
     490  % \label{eq:MB_cor+metric} 
    491491  f \to f + \frac{1}{e_1 e_2} \lt( v \pd[e_2]{i} - u \pd[e_1]{j} \rt) 
    492492\] 
     
    503503  \textbf{Vector invariant form of the momentum equations}: 
    504504  \begin{equation} 
    505     \label{eq:PE_dyn_vect} 
     505    \label{eq:MB_dyn_vect} 
    506506    \begin{split} 
    507     % \label{eq:PE_dyn_vect_u} 
     507    % \label{eq:MB_dyn_vect_u} 
    508508      \pd[u]{t} = &+ (\zeta + f) \, v - \frac{1}{2 e_1} \pd[]{i} (u^2 + v^2) 
    509509                   - \frac{1}{e_3} w \pd[u]{k} - \frac{1}{e_1} \pd[]{i} \lt( \frac{p_s + p_h}{\rho_o} \rt) \\ 
     
    516516\item 
    517517  \textbf{flux form of the momentum equations}: 
    518   % \label{eq:PE_dyn_flux} 
     518  % \label{eq:MB_dyn_flux} 
    519519  \begin{multline*} 
    520     % \label{eq:PE_dyn_flux_u} 
     520    % \label{eq:MB_dyn_flux_u} 
    521521    \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 \\ 
    522522                - \frac{1}{e_1 \; e_2} \lt( \pd[(e_2 \, u \, u)]{i} + \pd[(e_1 \, v \, u)]{j} \rt) \\ 
     
    525525  \end{multline*} 
    526526  \begin{multline*} 
    527     % \label{eq:PE_dyn_flux_v} 
     527    % \label{eq:MB_dyn_flux_v} 
    528528    \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 \\ 
    529529                - \frac{1}{e_1 \; e_2} \lt( \pd[(e_2 \, u \, v)]{i} + \pd[(e_1 \, v \, v)]{j} \rt) \\ 
     
    531531                + D_v^{\vect U} + F_v^{\vect U} 
    532532  \end{multline*} 
    533   where $\zeta$, the relative vorticity, is given by \autoref{eq:PE_curl_Uh} and $p_s$, the surface pressure, 
     533  where $\zeta$, the relative vorticity, is given by \autoref{eq:MB_curl_Uh} and $p_s$, the surface pressure, 
    534534  is given by: 
    535535  \[ 
    536   % \label{eq:PE_spg} 
     536  % \label{eq:MB_spg} 
    537537    p_s = \rho \,g \, \eta 
    538538  \] 
    539   and $\eta$ is the solution of \autoref{eq:PE_ssh}. 
     539  and $\eta$ is the solution of \autoref{eq:MB_ssh}. 
    540540 
    541541  The vertical velocity and the hydrostatic pressure are diagnosed from the following equations: 
    542542  \[ 
    543   % \label{eq:w_diag} 
     543  % \label{eq:MB_w_diag} 
    544544    \pd[w]{k} = - \chi \; e_3 \qquad 
    545   % \label{eq:hp_diag} 
     545  % \label{eq:MB_hp_diag} 
    546546    \pd[p_h]{k} = - \rho \; g \; e_3 
    547547  \] 
    548   where the divergence of the horizontal velocity, $\chi$ is given by \autoref{eq:PE_div_Uh}. 
     548  where the divergence of the horizontal velocity, $\chi$ is given by \autoref{eq:MB_div_Uh}. 
    549549 
    550550\item 
     
    562562 
    563563The expression of $\vect D^{U}$, $D^{S}$ and $D^{T}$ depends on the subgrid scale parameterisation used. 
    564 It will be defined in \autoref{eq:PE_zdf}. 
     564It will be defined in \autoref{eq:MB_zdf}. 
    565565The nature and formulation of $\vect F^{\vect U}$, $F^T$ and $F^S$, the surface forcing terms, 
    566566are discussed in \autoref{chap:SBC}. 
     
    572572% ================================================================ 
    573573\section{Curvilinear generalised vertical coordinate system} 
    574 \label{sec:PE_gco} 
     574\label{sec:MB_gco} 
    575575 
    576576The ocean domain presents a huge diversity of situation in the vertical. 
     
    596596introducing an arbitrary vertical coordinate : 
    597597\begin{equation} 
    598   \label{eq:PE_s} 
     598  \label{eq:MB_s} 
    599599  s = s(i,j,k,t) 
    600600\end{equation} 
    601601with the restriction that the above equation gives a single-valued monotonic relationship between $s$ and $k$, 
    602602when $i$, $j$ and $t$ are held fixed. 
    603 \autoref{eq:PE_s} is a transformation from the $(i,j,k,t)$ coordinate system with independent variables into 
     603\autoref{eq:MB_s} is a transformation from the $(i,j,k,t)$ coordinate system with independent variables into 
    604604the $(i,j,s,t)$ generalised coordinate system with $s$ depending on the other three variables through 
    605 \autoref{eq:PE_s}. 
     605\autoref{eq:MB_s}. 
    606606This so-called \textit{generalised vertical coordinate} \citep{kasahara_MWR74} is in fact 
    607607an Arbitrary Lagrangian--Eulerian (ALE) coordinate. 
     
    656656\subsection{\textit{S}-coordinate formulation} 
    657657 
    658 Starting from the set of equations established in \autoref{sec:PE_zco} for the special case $k = z$ and 
     658Starting from the set of equations established in \autoref{sec:MB_zco} for the special case $k = z$ and 
    659659thus $e_3 = 1$, we introduce an arbitrary vertical coordinate $s = s(i,j,k,t)$, 
    660660which includes $z$-, \zstar- and $\sigma$-coordinates as special cases 
    661661($s = z$, $s = \zstar$, and $s = \sigma = z / H$ or $ = z / \lt( H + \eta \rt)$, resp.). 
    662 A formal derivation of the transformed equations is given in \autoref{apdx:A}. 
     662A formal derivation of the transformed equations is given in \autoref{apdx:SCOORD}. 
    663663Let us define the vertical scale factor by $e_3 = \partial_s z$  ($e_3$ is now a function of $(i,j,k,t)$ ), 
    664664and the slopes in the $(i,j)$ directions between $s$- and $z$-surfaces by: 
    665665\begin{equation} 
    666   \label{eq:PE_sco_slope} 
     666  \label{eq:MB_sco_slope} 
    667667  \sigma_1 = \frac{1}{e_1} \; \lt. \pd[z]{i} \rt|_s \quad \text{and} \quad 
    668668  \sigma_2 = \frac{1}{e_2} \; \lt. \pd[z]{j} \rt|_s 
     
    671671relative to the moving $s$-surfaces and normal to them: 
    672672\[ 
    673   % \label{eq:PE_sco_w} 
     673  % \label{eq:MB_sco_w} 
    674674  \omega = w -  \, \lt. \pd[z]{t} \rt|_s - \sigma_1 \, u - \sigma_2 \, v 
    675675\] 
    676676 
    677 The equations solved by the ocean model \autoref{eq:PE} in $s$-coordinate can be written as follows 
    678 (see \autoref{sec:A_momentum}): 
     677The equations solved by the ocean model \autoref{eq:MB_PE} in $s$-coordinate can be written as follows 
     678(see \autoref{sec:SCOORD_momentum}): 
    679679 
    680680\begin{itemize} 
    681681\item \textbf{Vector invariant form of the momentum equation}: 
    682682  \begin{multline*} 
    683   % \label{eq:PE_sco_u_vector} 
     683  % \label{eq:MB_sco_u_vector} 
    684684    \pd[u]{t} = + (\zeta + f) \, v - \frac{1}{2 \, e_1} \pd[]{i} (u^2 + v^2) - \frac{1}{e_3} \omega \pd[u]{k} \\ 
    685685                - \frac{1}{e_1} \pd[]{i} \lt( \frac{p_s + p_h}{\rho_o} \rt) - g \frac{\rho}{\rho_o} \sigma_1 
     
    687687  \end{multline*} 
    688688  \begin{multline*} 
    689   % \label{eq:PE_sco_v_vector} 
     689  % \label{eq:MB_sco_v_vector} 
    690690    \pd[v]{t} = - (\zeta + f) \, u - \frac{1}{2 \, e_2} \pd[]{j}(u^2 + v^2) - \frac{1}{e_3} \omega \pd[v]{k} \\ 
    691691                - \frac{1}{e_2} \pd[]{j} \lt( \frac{p_s + p_h}{\rho_o} \rt) - g \frac{\rho}{\rho_o} \sigma_2 
     
    694694\item \textbf{Flux form of the momentum equation}: 
    695695  \begin{multline*} 
    696   % \label{eq:PE_sco_u_flux} 
     696  % \label{eq:MB_sco_u_flux} 
    697697    \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 \\ 
    698698                                       - \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) \\ 
     
    702702  \end{multline*} 
    703703  \begin{multline*} 
    704   % \label{eq:PE_sco_v_flux} 
     704  % \label{eq:MB_sco_v_flux} 
    705705    \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 \\ 
    706706                                       - \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) \\ 
     
    712712  and the hydrostatic pressure have the same expressions as in $z$-coordinates although 
    713713  they do not represent exactly the same quantities. 
    714   $\omega$ is provided by the continuity equation (see \autoref{apdx:A}): 
     714  $\omega$ is provided by the continuity equation (see \autoref{apdx:SCOORD}): 
    715715  \[ 
    716   % \label{eq:PE_sco_continuity} 
     716  % \label{eq:MB_sco_continuity} 
    717717    \pd[e_3]{t} + e_3 \; \chi + \pd[\omega]{s} = 0 \quad \text{with} \quad 
    718718    \chi = \frac{1}{e_1 e_2 e_3} \lt( \pd[(e_2 e_3 \, u)]{i} + \pd[(e_1 e_3 \, v)]{j} \rt) 
     
    720720\item \textit{tracer equations}: 
    721721  \begin{multline*} 
    722   % \label{eq:PE_sco_t} 
     722  % \label{eq:MB_sco_t} 
    723723    \frac{1}{e_3} \pd[(e_3 \, T)]{t} = - \frac{1}{e_1 e_2 e_3} \lt(   \pd[(e_2 e_3 \, u \, T)]{i} 
    724724                                                                    + \pd[(e_1 e_3 \, v \, T)]{j} \rt) \\ 
     
    726726  \end{multline*} 
    727727  \begin{multline} 
    728   % \label{eq:PE_sco_s} 
     728  % \label{eq:MB_sco_s} 
    729729    \frac{1}{e_3} \pd[(e_3 \, S)]{t} = - \frac{1}{e_1 e_2 e_3} \lt(   \pd[(e_2 e_3 \, u \, S)]{i} 
    730730                                                                    + \pd[(e_1 e_3 \, v \, S)]{j} \rt) \\ 
     
    745745% ------------------------------------------------------------------------------------------------------------- 
    746746\subsection{Curvilinear \zstar-coordinate system} 
    747 \label{subsec:PE_zco_star} 
     747\label{subsec:MB_zco_star} 
    748748 
    749749%>>>>>>>>>>>>>>>>>>>>>>>>>>>> 
     
    752752    \includegraphics[width=\textwidth]{Fig_z_zstar} 
    753753    \caption{ 
    754       \protect\label{fig:z_zstar} 
     754      \protect\label{fig:MB_z_zstar} 
    755755      (a) $z$-coordinate in linear free-surface case ; 
    756756      (b) $z$-coordinate in non-linear free surface case ; 
     
    771771as in the $z$-coordinate formulation, but is equally distributed over the full water column. 
    772772Thus vertical levels naturally follow sea-surface variations, with a linear attenuation with depth, 
    773 as illustrated by \autoref{fig:z_zstar}. 
    774 Note that with a flat bottom, such as in \autoref{fig:z_zstar}, the bottom-following $z$ coordinate and \zstar are equivalent. 
     773as illustrated by \autoref{fig:MB_z_zstar}. 
     774Note that with a flat bottom, such as in \autoref{fig:MB_z_zstar}, the bottom-following $z$ coordinate and \zstar are equivalent. 
    775775The definition and modified oceanic equations for the rescaled vertical coordinate \zstar, 
    776776including the treatment of fresh-water flux at the surface, are detailed in Adcroft and Campin (2004). 
     
    778778The position (\zstar) and vertical discretization (\zstar) are expressed as: 
    779779\[ 
    780   % \label{eq:PE_z-star} 
     780  % \label{eq:MB_z-star} 
    781781  H + \zstar = (H + z)  / r \quad \text{and}  \quad \delta \zstar 
    782782              = \delta z / r \quad \text{with} \quad r 
     
    785785Simple re-organisation of the above expressions gives 
    786786\[ 
    787   % \label{eq:PE_zstar_2} 
     787  % \label{eq:MB_zstar_2} 
    788788  \zstar = H \lt( \frac{z - \eta}{H + \eta} \rt) . 
    789789\] 
     
    806806it is clear that surfaces constant \zstar are very similar to the depth surfaces. 
    807807These properties greatly reduce difficulties of computing the horizontal pressure gradient relative to 
    808 terrain following sigma models discussed in \autoref{subsec:PE_sco}. 
     808terrain following sigma models discussed in \autoref{subsec:MB_sco}. 
    809809Additionally, since $\zstar = z$ when $\eta = 0$, 
    810810no flow is spontaneously generated in an unforced ocean starting from rest, regardless the bottom topography. 
     
    839839% ------------------------------------------------------------------------------------------------------------- 
    840840\subsection{Curvilinear terrain-following \textit{s}--coordinate} 
    841 \label{subsec:PE_sco} 
     841\label{subsec:MB_sco} 
    842842 
    843843% ------------------------------------------------------------------------------------------------------------- 
     
    851851For example, the topographic $\beta$-effect is usually larger than the planetary one along continental slopes. 
    852852Topographic Rossby waves can be excited and can interact with the mean current. 
    853 In the $z$-coordinate system presented in the previous section (\autoref{sec:PE_zco}), 
     853In the $z$-coordinate system presented in the previous section (\autoref{sec:MB_zco}), 
    854854$z$-surfaces are geopotential surfaces. 
    855855The bottom topography is discretised by steps. 
     
    875875The main two problems come from the truncation error in the horizontal pressure gradient and 
    876876a possibly increased diapycnal diffusion. 
    877 The horizontal pressure force in $s$-coordinate consists of two terms (see \autoref{apdx:A}), 
     877The horizontal pressure force in $s$-coordinate consists of two terms (see \autoref{apdx:SCOORD}), 
    878878 
    879879\begin{equation} 
    880   \label{eq:PE_p_sco} 
     880  \label{eq:MB_p_sco} 
    881881  \nabla p |_z = \nabla p |_s - \frac{1}{e_3} \pd[p]{s} \nabla z |_s 
    882882\end{equation} 
    883883 
    884 The second term in \autoref{eq:PE_p_sco} depends on the tilt of the coordinate surface and 
     884The second term in \autoref{eq:MB_p_sco} depends on the tilt of the coordinate surface and 
    885885leads to a truncation error that is not present in a $z$-model. 
    886886In the special case of a $\sigma$-coordinate (i.e. a depth-normalised coordinate system $\sigma = z/H$), 
     
    898898an envelope topography is defined in $s$-coordinate on which a full or 
    899899partial step bottom topography is then applied in order to adjust the model depth to the observed one 
    900 (see \autoref{sec:DOM_zgr}. 
     900(see \autoref{subsec:DOM_zgr}. 
    901901 
    902902For numerical reasons a minimum of diffusion is required along the coordinate surfaces of 
     
    915915the strongly stratified portion of the water column (\ie\ the main thermocline) \citep{madec.delecluse.ea_JPO96}. 
    916916An alternate solution consists of rotating the lateral diffusive tensor to geopotential or to isoneutral surfaces 
    917 (see \autoref{subsec:PE_ldf}). 
     917(see \autoref{subsec:MB_ldf}). 
    918918Unfortunately, the slope of isoneutral surfaces relative to the $s$-surfaces can very large, 
    919919strongly exceeding the stability limit of such a operator when it is discretized (see \autoref{chap:LDF}). 
     
    928928% ------------------------------------------------------------------------------------------------------------- 
    929929\subsection{\texorpdfstring{Curvilinear \ztilde-coordinate}{}} 
    930 \label{subsec:PE_zco_tilde} 
     930\label{subsec:MB_zco_tilde} 
    931931 
    932932The \ztilde -coordinate has been developed by \citet{leclair.madec_OM11}. 
     
    941941% ================================================================ 
    942942\section{Subgrid scale physics} 
    943 \label{sec:PE_zdf_ldf} 
     943\label{sec:MB_zdf_ldf} 
    944944 
    945945The hydrostatic primitive equations describe the behaviour of a geophysical fluid at space and time scales larger than 
     
    958958The control exerted by gravity on the flow induces a strong anisotropy between the lateral and vertical motions. 
    959959Therefore subgrid-scale physics \textbf{D}$^{\vect U}$, $D^{S}$ and $D^{T}$  in 
    960 \autoref{eq:PE_dyn}, \autoref{eq:PE_tra_T} and \autoref{eq:PE_tra_S} are divided into 
     960\autoref{eq:MB_PE_dyn}, \autoref{eq:MB_PE_tra_T} and \autoref{eq:MB_PE_tra_S} are divided into 
    961961a lateral part \textbf{D}$^{l \vect U}$, $D^{l S}$ and $D^{l T}$ and 
    962962a vertical part \textbf{D}$^{v \vect U}$, $D^{v S}$ and $D^{v T}$. 
     
    967967% ------------------------------------------------------------------------------------------------------------- 
    968968\subsection{Vertical subgrid scale physics} 
    969 \label{subsec:PE_zdf} 
     969\label{subsec:MB_zdf} 
    970970 
    971971The model resolution is always larger than the scale at which the major sources of vertical turbulence occur 
     
    981981The resulting vertical momentum and tracer diffusive operators are of second order: 
    982982\begin{equation} 
    983   \label{eq:PE_zdf} 
     983  \label{eq:MB_zdf} 
    984984  \begin{gathered} 
    985985    \vect D^{v \vect U} = \pd[]{z} \lt( A^{vm} \pd[\vect U_h]{z} \rt) \ , \\ 
     
    10011001% ------------------------------------------------------------------------------------------------------------- 
    10021002\subsection{Formulation of the lateral diffusive and viscous operators} 
    1003 \label{subsec:PE_ldf} 
     1003\label{subsec:MB_ldf} 
    10041004 
    10051005Lateral turbulence can be roughly divided into a mesoscale turbulence associated with eddies 
     
    10551055\subsubsection{Lateral laplacian tracer diffusive operator} 
    10561056 
    1057 The lateral Laplacian tracer diffusive operator is defined by (see \autoref{apdx:B}): 
     1057The lateral Laplacian tracer diffusive operator is defined by (see \autoref{apdx:DIFFOPERS}): 
    10581058\begin{equation} 
    1059   \label{eq:PE_iso_tensor} 
     1059  \label{eq:MB_iso_tensor} 
    10601060  D^{lT} = \nabla \vect . \lt( A^{lT} \; \Re \; \nabla T \rt) \quad \text{with} \quad 
    10611061  \Re = 
     
    10681068where $r_1$ and $r_2$ are the slopes between the surface along which the diffusive operator acts and 
    10691069the model level (\eg\ $z$- or $s$-surfaces). 
    1070 Note that the formulation \autoref{eq:PE_iso_tensor} is exact for 
     1070Note that the formulation \autoref{eq:MB_iso_tensor} is exact for 
    10711071the rotation between geopotential and $s$-surfaces, 
    10721072while it is only an approximation for the rotation between isoneutral and $z$- or $s$-surfaces. 
    1073 Indeed, in the latter case, two assumptions are made to simplify \autoref{eq:PE_iso_tensor} \citep{cox_OM87}. 
     1073Indeed, in the latter case, two assumptions are made to simplify \autoref{eq:MB_iso_tensor} \citep{cox_OM87}. 
    10741074First, the horizontal contribution of the dianeutral mixing is neglected since the ratio between iso and 
    10751075dia-neutral diffusive coefficients is known to be several orders of magnitude smaller than unity. 
    10761076Second, the two isoneutral directions of diffusion are assumed to be independent since 
    1077 the slopes are generally less than $10^{-2}$ in the ocean (see \autoref{apdx:B}). 
     1077the slopes are generally less than $10^{-2}$ in the ocean (see \autoref{apdx:DIFFOPERS}). 
    10781078 
    10791079For \textit{iso-level} diffusion, $r_1$ and $r_2 $ are zero. 
     
    10821082For \textit{geopotential} diffusion, 
    10831083$r_1$ and $r_2 $ are the slopes between the geopotential and computational surfaces: 
    1084 they are equal to $\sigma_1$ and $\sigma_2$, respectively (see \autoref{eq:PE_sco_slope}). 
     1084they are equal to $\sigma_1$ and $\sigma_2$, respectively (see \autoref{eq:MB_sco_slope}). 
    10851085 
    10861086For \textit{isoneutral} diffusion $r_1$ and $r_2$ are the slopes between the isoneutral and computational surfaces. 
     
    10881088In $z$-coordinates: 
    10891089\begin{equation} 
    1090   \label{eq:PE_iso_slopes} 
     1090  \label{eq:MB_iso_slopes} 
    10911091  r_1 = \frac{e_3}{e_1} \lt( \pd[\rho]{i} \rt) \lt( \pd[\rho]{k} \rt)^{-1} \quad 
    10921092  r_2 = \frac{e_3}{e_2} \lt( \pd[\rho]{j} \rt) \lt( \pd[\rho]{k} \rt)^{-1} 
     
    10991099an additional tracer advection is introduced in combination with the isoneutral diffusion of tracers: 
    11001100\[ 
    1101   % \label{eq:PE_iso+eiv} 
     1101  % \label{eq:MB_iso+eiv} 
    11021102  D^{lT} = \nabla \cdot \lt( A^{lT} \; \Re \; \nabla T \rt) + \nabla \cdot \lt( \vect U^\ast \, T \rt) 
    11031103\] 
     
    11051105eddy-induced transport velocity. This velocity field is defined by: 
    11061106\begin{gather} 
    1107   % \label{eq:PE_eiv} 
     1107  % \label{eq:MB_eiv} 
    11081108  u^\ast =   \frac{1}{e_3}            \pd[]{k} \lt( A^{eiv} \;        \tilde{r}_1 \rt) \\ 
    11091109  v^\ast =   \frac{1}{e_3}            \pd[]{k} \lt( A^{eiv} \;        \tilde{r}_2 \rt) \\ 
     
    11161116Their values are thus independent of the vertical coordinate, but their expression depends on the coordinate: 
    11171117\begin{align} 
    1118   \label{eq:PE_slopes_eiv} 
     1118  \label{eq:MB_slopes_eiv} 
    11191119  \tilde{r}_n = 
    11201120    \begin{cases} 
     
    11341134The lateral bilaplacian tracer diffusive operator is defined by: 
    11351135\[ 
    1136   % \label{eq:PE_bilapT} 
     1136  % \label{eq:MB_bilapT} 
    11371137  D^{lT}= - \Delta \; (\Delta T) \quad \text{where} \quad 
    11381138  \Delta \bullet = \nabla \lt( \sqrt{B^{lT}} \; \Re \; \nabla \bullet \rt) 
    11391139\] 
    1140 It is the Laplacian operator given by \autoref{eq:PE_iso_tensor} applied twice with 
     1140It is the Laplacian operator given by \autoref{eq:MB_iso_tensor} applied twice with 
    11411141the harmonic eddy diffusion coefficient set to the square root of the biharmonic one. 
    11421142 
     
    11441144 
    11451145The Laplacian momentum diffusive operator along $z$- or $s$-surfaces is found by 
    1146 applying \autoref{eq:PE_lap_vector} to the horizontal velocity vector (see \autoref{apdx:B}): 
     1146applying \autoref{eq:MB_lap_vector} to the horizontal velocity vector (see \autoref{apdx:DIFFOPERS}): 
    11471147\begin{align*} 
    1148   % \label{eq:PE_lapU} 
     1148  % \label{eq:MB_lapU} 
    11491149  \vect D^{l \vect U} &=   \nabla_h        \big( A^{lm}    \chi             \big) 
    11501150                         - \nabla_h \times \big( A^{lm} \, \zeta \; \vect k \big) \\ 
     
    11561156 
    11571157Such a formulation ensures a complete separation between the vorticity and horizontal divergence fields 
    1158 (see \autoref{apdx:C}). 
     1158(see \autoref{apdx:INVARIANTS}). 
    11591159Unfortunately, it is only available in \textit{iso-level} direction. 
    11601160When a rotation is required 
     
    11621162the $u$ and $v$-fields are considered as independent scalar fields, so that the diffusive operator is given by: 
    11631163\begin{gather*} 
    1164   % \label{eq:PE_lapU_iso} 
     1164  % \label{eq:MB_lapU_iso} 
    11651165    D_u^{l \vect U} = \nabla . \lt( A^{lm} \; \Re \; \nabla u \rt) \\ 
    11661166    D_v^{l \vect U} = \nabla . \lt( A^{lm} \; \Re \; \nabla v \rt) 
    11671167\end{gather*} 
    1168 where $\Re$ is given by \autoref{eq:PE_iso_tensor}. 
     1168where $\Re$ is given by \autoref{eq:MB_iso_tensor}. 
    11691169It is the same expression as those used for diffusive operator on tracers. 
    11701170It must be emphasised that such a formulation is only exact in a Cartesian coordinate system, 
  • NEMO/trunk/doc/latex/NEMO/subfiles/chap_model_basics_zstar.tex

    r11537 r11543  
    4040the surface height, it is clear that surfaces constant $z^\star$ are very similar to the depth surfaces. 
    4141These properties greatly reduce difficulties of computing the horizontal pressure gradient relative to 
    42 terrain following sigma models discussed in \autoref{subsec:PE_sco}. 
     42terrain following sigma models discussed in \autoref{subsec:MB_sco}. 
    4343Additionally, since $z^\star$ when $\eta = 0$, no flow is spontaneously generated in 
    4444an unforced ocean starting from rest, regardless the bottom topography. 
     
    8181%------------------------------------------------------------------------------------------------------------ 
    8282Options are defined through the \nam{\_dynspg} namelist variables. 
    83 The surface pressure gradient term is related to the representation of the free surface (\autoref{sec:PE_hor_pg}). 
     83The surface pressure gradient term is related to the representation of the free surface (\autoref{sec:MB_hor_pg}). 
    8484The main distinction is between the fixed volume case (linear free surface or rigid lid) and 
    8585the variable volume case (nonlinear free surface, \key{vvl} is active). 
    86 In the linear free surface case (\autoref{subsec:PE_free_surface}) and rigid lid (\autoref{PE_rigid_lid}), 
     86In the linear free surface case (\autoref{subsec:MB_free_surface}) and rigid lid (\autoref{PE_rigid_lid}), 
    8787the vertical scale factors $e_{3}$ are fixed in time, 
    88 while in the nonlinear case (\autoref{subsec:PE_free_surface}) they are time-dependent. 
     88while in the nonlinear case (\autoref{subsec:MB_free_surface}) they are time-dependent. 
    8989With both linear and nonlinear free surface, external gravity waves are allowed in the equations, 
    9090which imposes a very small time step when an explicit time stepping is used. 
    9191Two methods are proposed to allow a longer time step for the three-dimensional equations: 
    92 the filtered free surface, which is a modification of the continuous equations %(see \autoref{eq:PE_flt?}), 
     92the filtered free surface, which is a modification of the continuous equations %(see \autoref{eq:MB_flt?}), 
    9393and the split-explicit free surface described below. 
    9494The extra term introduced in the filtered method is calculated implicitly, 
     
    116116and $\rho_w =1,000\,Kg.m^{-3}$ is the volumic mass of pure water. 
    117117The sea-surface height is evaluated using a leapfrog scheme in combination with an Asselin time filter, 
    118 (\ie\ the velocity appearing in (\autoref{eq:dynspg_ssh}) is centred in time (\textit{now} velocity). 
     118(\ie\ the velocity appearing in (\autoref{eq:DYN_spg_ssh}) is centred in time (\textit{now} velocity). 
    119119 
    120120The surface pressure gradient, also evaluated using a leap-frog scheme, is then simply given by: 
     
    130130 
    131131Consistent with the linearization, a $\left. \rho \right|_{k=1} / \rho_o$ factor is omitted in 
    132 (\autoref{eq:dynspg_exp}). 
     132(\autoref{eq:DYN_spg_exp}). 
    133133 
    134134%------------------------------------------------------------- 
     
    316316This option is presented in a report \citep{levier.treguier.ea_rpt07} available on the \NEMO\ web site. 
    317317The three time-stepping methods (explicit, split-explicit and filtered) are the same as in 
    318 \autoref{DYN_spg_linear} except that the ocean depth is now time-dependent. 
     318\autoref{?:DYN_spg_linear?} except that the ocean depth is now time-dependent. 
    319319In particular, this means that in filtered case, the matrix to be inverted has to be recomputed at each time-step. 
    320320 
  • NEMO/trunk/doc/latex/NEMO/subfiles/chap_time_domain.tex

    r11537 r11543  
    66% Chapter 2 ——— Time Domain (step.F90) 
    77% ================================================================ 
    8 \chapter{Time Domain (STP)} 
    9 \label{chap:STP} 
     8\chapter{Time Domain} 
     9\label{chap:TD} 
    1010\chaptertoc 
    1111 
     
    1919\newpage 
    2020 
    21 Having defined the continuous equations in \autoref{chap:PE}, we need now to choose a time discretization, 
     21Having defined the continuous equations in \autoref{chap:MB}, we need now to choose a time discretization, 
    2222a key feature of an ocean model as it exerts a strong influence on the structure of the computer code 
    2323(\ie\ on its flowchart). 
     
    2929% ================================================================ 
    3030\section{Time stepping environment} 
    31 \label{sec:STP_environment} 
     31\label{sec:TD_environment} 
    3232 
    3333The time stepping used in \NEMO\ is a three level scheme that can be represented as follows: 
    3434\begin{equation} 
    35   \label{eq:STP} 
     35  \label{eq:TD} 
    3636  x^{t + \rdt} = x^{t - \rdt} + 2 \, \rdt \ \text{RHS}_x^{t - \rdt, \, t, \, t + \rdt} 
    3737\end{equation} 
     
    5252The third array, although referred to as $x_a$ (after) in the code, 
    5353is usually not the variable at the after time step; 
    54 but rather it is used to store the time derivative (RHS in \autoref{eq:STP}) prior to time-stepping the equation. 
     54but rather it is used to store the time derivative (RHS in \autoref{eq:TD}) prior to time-stepping the equation. 
    5555The time stepping itself is performed once at each time step where implicit vertical diffusion is computed, \ie\ in the \mdl{trazdf} and \mdl{dynzdf} modules. 
    5656 
     
    5959% ------------------------------------------------------------------------------------------------------------- 
    6060\section{Non-diffusive part --- Leapfrog scheme} 
    61 \label{sec:STP_leap_frog} 
     61\label{sec:TD_leap_frog} 
    6262 
    6363The time stepping used for processes other than diffusion is the well-known leapfrog scheme 
    6464\citep{mesinger.arakawa_bk76}. 
    6565This scheme is widely used for advection processes in low-viscosity fluids. 
    66 It is a time centred scheme, \ie\ the RHS in \autoref{eq:STP} is evaluated at time step $t$, the now time step. 
     66It is a time centred scheme, \ie\ the RHS in \autoref{eq:TD} is evaluated at time step $t$, the now time step. 
    6767It may be used for momentum and tracer advection, pressure gradient, and Coriolis terms, 
    6868but not for diffusion terms. 
     
    8181is a kind of laplacian diffusion in time that mixes odd and even time steps: 
    8282\begin{equation} 
    83   \label{eq:STP_asselin} 
     83  \label{eq:TD_asselin} 
    8484  x_F^t = x^t + \gamma \, \lt[ x_F^{t - \rdt} - 2 x^t + x^{t + \rdt} \rt] 
    8585\end{equation} 
    8686where the subscript $F$ denotes filtered values and $\gamma$ is the Asselin coefficient. 
    8787$\gamma$ is initialized as \np{rn\_atfp} (namelist parameter). 
    88 Its default value is \np{rn\_atfp}\forcode{=10.e-3} (see \autoref{sec:STP_mLF}), 
     88Its default value is \np{rn\_atfp}\forcode{ = 10.e-3} (see \autoref{sec:TD_mLF}), 
    8989causing only a weak dissipation of high frequency motions (\citep{farge-coulombier_phd87}). 
    9090The addition of a time filter degrades the accuracy of the calculation from second to first order. 
     
    102102% ------------------------------------------------------------------------------------------------------------- 
    103103\section{Diffusive part --- Forward or backward scheme} 
    104 \label{sec:STP_forward_imp} 
     104\label{sec:TD_forward_imp} 
    105105 
    106106The leapfrog differencing scheme is unsuitable for the representation of diffusion and damping processes. 
     
    108108(when present, see \autoref{sec:TRA_dmp}), a forward time differencing scheme is used : 
    109109\[ 
    110   %\label{eq:STP_euler} 
     110  %\label{eq:TD_euler} 
    111111  x^{t + \rdt} = x^{t - \rdt} + 2 \, \rdt \ D_x^{t - \rdt} 
    112112\] 
     
    115115The conditions for stability of second and fourth order horizontal diffusion schemes are \citep{griffies_bk04}: 
    116116\begin{equation} 
    117   \label{eq:STP_euler_stability} 
     117  \label{eq:TD_euler_stability} 
    118118  A^h < 
    119119  \begin{cases} 
     
    123123\end{equation} 
    124124where $e$ is the smallest grid size in the two horizontal directions and $A^h$ is the mixing coefficient. 
    125 The linear constraint \autoref{eq:STP_euler_stability} is a necessary condition, but not sufficient. 
     125The linear constraint \autoref{eq:TD_euler_stability} is a necessary condition, but not sufficient. 
    126126If it is not satisfied, even mildly, then the model soon becomes wildly unstable. 
    127127The instability can be removed by either reducing the length of the time steps or reducing the mixing coefficient. 
     
    131131backward (or implicit) time differencing scheme is used. This scheme is unconditionally stable but diffusive and can be written as follows: 
    132132\begin{equation} 
    133   \label{eq:STP_imp} 
     133  \label{eq:TD_imp} 
    134134  x^{t + \rdt} = x^{t - \rdt} + 2 \, \rdt \ \text{RHS}_x^{t + \rdt} 
    135135\end{equation} 
     
    141141This scheme is rather time consuming since it requires a matrix inversion. For example, the finite difference approximation of the temperature equation is: 
    142142\[ 
    143   % \label{eq:STP_imp_zdf} 
     143  % \label{eq:TD_imp_zdf} 
    144144  \frac{T(k)^{t + 1} - T(k)^{t - 1}}{2 \; \rdt} 
    145145  \equiv 
     
    147147\] 
    148148where RHS is the right hand side of the equation except for the vertical diffusion term. 
    149 We rewrite \autoref{eq:STP_imp} as: 
    150 \begin{equation} 
    151   \label{eq:STP_imp_mat} 
     149We rewrite \autoref{eq:TD_imp} as: 
     150\begin{equation} 
     151  \label{eq:TD_imp_mat} 
    152152  -c(k + 1) \; T^{t + 1}(k + 1) + d(k) \; T^{t + 1}(k) - \; c(k) \; T^{t + 1}(k - 1) \equiv b(k) 
    153153\end{equation} 
     
    159159\end{align*} 
    160160 
    161 \autoref{eq:STP_imp_mat} is a linear system of equations with an associated matrix which is tridiagonal. 
     161\autoref{eq:TD_imp_mat} is a linear system of equations with an associated matrix which is tridiagonal. 
    162162Moreover, 
    163163$c(k)$ and $d(k)$ are positive and the diagonal term is greater than the sum of the two extra-diagonal terms, 
     
    169169% ------------------------------------------------------------------------------------------------------------- 
    170170\section{Surface pressure gradient} 
    171 \label{sec:STP_spg_ts} 
     171\label{sec:TD_spg_ts} 
    172172 
    173173The leapfrog environment supports a centred in time computation of the surface pressure, \ie\ evaluated 
     
    177177(\np{ln\_dynspg\_ts}\forcode{=.true.}) in which barotropic and baroclinic dynamical equations are solved separately with ad-hoc 
    178178time steps. The use of the time-splitting (in combination with non-linear free surface) imposes some constraints on the design of 
    179 the overall flowchart, in particular to ensure exact tracer conservation (see \autoref{fig:TimeStep_flowchart}). 
     179the overall flowchart, in particular to ensure exact tracer conservation (see \autoref{fig:TD_TimeStep_flowchart}). 
    180180 
    181181Compared to the former use of the filtered free surface in \NEMO\ v3.6 (\citet{roullet.madec_JGR00}), the use of a split-explicit free surface is advantageous 
     
    189189    \includegraphics[width=\textwidth]{Fig_TimeStepping_flowchart_v4} 
    190190    \caption{ 
    191       \protect\label{fig:TimeStep_flowchart} 
     191      \protect\label{fig:TD_TimeStep_flowchart} 
    192192      Sketch of the leapfrog time stepping sequence in \NEMO\ with split-explicit free surface. The latter combined 
    193193       with non-linear free surface requires the dynamical tendency being updated prior tracers tendency to ensure 
     
    205205% ------------------------------------------------------------------------------------------------------------- 
    206206\section{Modified Leapfrog -- Asselin filter scheme} 
    207 \label{sec:STP_mLF} 
     207\label{sec:TD_mLF} 
    208208 
    209209Significant changes have been introduced by \cite{leclair.madec_OM09} in the LF-RA scheme in order to 
     
    214214\ie\ it is time-stepped over a $2 \rdt$ period: 
    215215$x^t = x^t + 2 \rdt Q^t$ where $Q$ is the forcing applied to $x$, 
    216 and the time filter is given by \autoref{eq:STP_asselin} so that $Q$ is redistributed over several time step. 
     216and the time filter is given by \autoref{eq:TD_asselin} so that $Q$ is redistributed over several time step. 
    217217In the modified LF-RA environment, these two formulations have been replaced by: 
    218218\begin{gather} 
    219   \label{eq:STP_forcing} 
     219  \label{eq:TD_forcing} 
    220220  x^{t + \rdt} = x^{t - \rdt} + \rdt \lt( Q^{t - \rdt / 2} + Q^{t + \rdt / 2} \rt)  \\ 
    221   \label{eq:STP_RA} 
     221  \label{eq:TD_RA} 
    222222  x_F^t       = x^t + \gamma \, \lt( x_F^{t - \rdt} - 2 x^t + x^{t + \rdt} \rt) 
    223223                    - \gamma \, \rdt \, \lt( Q^{t + \rdt / 2} - Q^{t - \rdt / 2} \rt) 
    224224\end{gather} 
    225 The change in the forcing formulation given by \autoref{eq:STP_forcing} (see \autoref{fig:MLF_forcing}) 
     225The change in the forcing formulation given by \autoref{eq:TD_forcing} (see \autoref{fig:TD_MLF_forcing}) 
    226226has a significant effect: 
    227227the forcing term no longer excites the divergence of odd and even time steps \citep{leclair.madec_OM09}. 
     
    231231Indeed, time filtering is no longer required on the forcing part. 
    232232The influence of the Asselin filter on the forcing is explicitly removed by adding a new term in the filter 
    233 (last term in \autoref{eq:STP_RA} compared to \autoref{eq:STP_asselin}). 
     233(last term in \autoref{eq:TD_RA} compared to \autoref{eq:TD_asselin}). 
    234234Since the filtering of the forcing was the source of non-conservation in the classical LF-RA scheme, 
    235235the modified formulation becomes conservative \citep{leclair.madec_OM09}. 
    236236Second, the LF-RA becomes a truly quasi -second order scheme. 
    237 Indeed, \autoref{eq:STP_forcing} used in combination with a careful treatment of static instability 
     237Indeed, \autoref{eq:TD_forcing} used in combination with a careful treatment of static instability 
    238238(\autoref{subsec:ZDF_evd}) and of the TKE physics (\autoref{subsec:ZDF_tke_ene}) 
    239239(the two other main sources of time step divergence), 
     
    242242Note that the forcing is now provided at the middle of a time step: 
    243243$Q^{t + \rdt / 2}$ is the forcing applied over the $[t,t + \rdt]$ time interval. 
    244 This and the change in the time filter, \autoref{eq:STP_RA}, 
     244This and the change in the time filter, \autoref{eq:TD_RA}, 
    245245allows for an exact evaluation of the contribution due to the forcing term between any two time steps, 
    246246even if separated by only $\rdt$ since the time filter is no longer applied to the forcing term. 
     
    251251    \includegraphics[width=\textwidth]{Fig_MLF_forcing} 
    252252    \caption{ 
    253       \protect\label{fig:MLF_forcing} 
     253      \protect\label{fig:TD_MLF_forcing} 
    254254      Illustration of forcing integration methods. 
    255255      (top) ''Traditional'' formulation: 
     
    268268% ------------------------------------------------------------------------------------------------------------- 
    269269\section{Start/Restart strategy} 
    270 \label{sec:STP_rst} 
     270\label{sec:TD_rst} 
    271271 
    272272%--------------------------------------------namrun------------------------------------------- 
     
    277277(Euler time integration): 
    278278\[ 
    279   % \label{eq:DOM_euler} 
     279  % \label{eq:TD_DOM_euler} 
    280280  x^1 = x^0 + \rdt \ \text{RHS}^0 
    281281\] 
    282 This is done simply by keeping the leapfrog environment (\ie\ the \autoref{eq:STP} three level time stepping) but 
     282This is done simply by keeping the leapfrog environment (\ie\ the \autoref{eq:TD} three level time stepping) but 
    283283setting all $x^0$ (\textit{before}) and $x^1$ (\textit{now}) fields equal at the first time step and 
    284284using half the value of a leapfrog time step ($2 \rdt$). 
     
    314314% ------------------------------------------------------------------------------------------------------------- 
    315315\subsection{Time domain} 
    316 \label{subsec:STP_time} 
     316\label{subsec:TD_time} 
    317317%--------------------------------------------namrun------------------------------------------- 
    318318 
  • NEMO/trunk/doc/latex/global/coding_rules.tex

    r11515 r11543  
    11 
    22\chapter{Coding Rules} 
    3 \label{apdx:coding} 
     3\label{apdx:CODING} 
    44 
    55\chaptertoc 
  • NEMO/trunk/doc/latex/global/document.tex

    r11524 r11543  
    1111 
    1212%% Document layout 
    13 \documentclass[draft, fontsize = 10pt, 
     13\documentclass[fontsize = 10pt, 
    1414twoside = semi, abstract = on, 
    1515open = right]{scrreprt} 
     
    4646\input{../../global/info_page} 
    4747 
     48\listoffigures \listoftables %\listoflistings   %% \listoflistings not working 
     49 
    4850\clearpage 
    4951 
    5052\pagenumbering{roman} 
     53\ofoot[]{\engine\ Reference Manual} \ifoot[]{\pagemark} 
     54 
    5155\input{introduction} 
    5256 
    5357%% Table of Contents 
    5458\tableofcontents 
    55 \listoffigures \listoftables \listoflistings 
    5659 
    5760\clearpage 
     
    7376 
    7477\appendix   %% Chapter numbering with letters by now 
     78\lohead{Apdx\ \thechapter\ \leftmark} 
    7579\include{appendices} 
    7680 
     
    7882\input{../../global/coding_rules} 
    7983 
     84\clearpage 
    8085 
    8186%% Backmatter 
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