Changeset 11596


Ignore:
Timestamp:
2019-09-25T19:06:37+02:00 (12 months ago)
Author:
nicolasmartin
Message:

Application of some coding rules

  • Replace comments before sectioning cmds by a single line of 100 characters long to display when every line should break
  • Replace multi blank lines by one single blank line
  • For list environment, put \item, label and content on the same line
  • Remove \newpage and comments line around figure envs
Location:
NEMO/trunk/doc/latex/NEMO/subfiles
Files:
24 edited

Legend:

Unmodified
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  • NEMO/trunk/doc/latex/NEMO/subfiles/apdx_DOMAINcfg.tex

    r11584 r11596  
    22 
    33\begin{document} 
    4 % ================================================================ 
    5 % Appendix DOMAINcfg : A brief guide to the DOMAINcfg tool 
    6 % ================================================================ 
    74\chapter{A brief guide to the DOMAINcfg tool} 
    85\label{apdx:DOMCFG} 
     
    1916\end{figure} 
    2017 
    21 \newpage 
    22  
    2318This appendix briefly describes some of the options available in the 
    2419\forcode{DOMAINcfg} tool mentioned in \autoref{chap:DOM}. 
     
    3530of those described elsewhere in this manual. 
    3631 
    37 % ------------------------------------------------------------------------------------------------------------- 
    38 %        Choice of horizontal grid 
    39 % ------------------------------------------------------------------------------------------------------------- 
    4032\section{Choice of horizontal grid} 
    4133\label{sec:DOMCFG_hor} 
     
    5547 
    5648\begin{description} 
    57  \item[{\np{jphgr_mesh}{jphgr\_mesh}=0}]  The most general curvilinear orthogonal grids. 
     49 \item [{\np{jphgr_mesh}{jphgr\_mesh}=0}]  The most general curvilinear orthogonal grids. 
    5850  The coordinates and their first derivatives with respect to $i$ and $j$ are provided 
    5951  in a input file (\ifile{coordinates}), read in \rou{hgr\_read} subroutine of the domhgr module. 
    6052  This is now the only option available within \NEMO\ itself from v4.0 onwards. 
    61 \item[{\np{jphgr_mesh}{jphgr\_mesh}=1 to 5}] A few simple analytical grids are provided (see below). 
     53\item [{\np{jphgr_mesh}{jphgr\_mesh}=1 to 5}] A few simple analytical grids are provided (see below). 
    6254  For other analytical grids, the \mdl{domhgr} module (\texttt{DOMAINcfg} variant) must be 
    6355  modified by the user. In most cases, modifying the \mdl{usrdef\_hgr} module of \NEMO\ is 
     
    10294(and the number of grid points). 
    10395 
    104 % ------------------------------------------------------------------------------------------------------------- 
    105 %        vertical reference coordinate transformation 
    106 % ------------------------------------------------------------------------------------------------------------- 
    10796\section{Vertical grid} 
    10897\label{sec:DOMCFG_vert} 
     
    111100\label{sec:DOMCFG_zref} 
    112101 
    113 %>>>>>>>>>>>>>>>>>>>>>>>>>>>> 
    114102\begin{figure}[!tb] 
    115103  \centering 
     
    121109  \label{fig:DOMCFG_zgr} 
    122110\end{figure} 
    123 %>>>>>>>>>>>>>>>>>>>>>>>>>>>> 
    124111 
    125112The reference coordinate transformation $z_0(k)$ defines the arrays $gdept_0$ and 
     
    210197999999$., in \nam{cfg}{cfg} namelist, and specifies instead the four following parameters: 
    211198\begin{itemize} 
    212 \item 
    213   \np{ppacr}{ppacr}~$= h_{cr}$: stretching factor (nondimensional). 
     199\item \np{ppacr}{ppacr}~$= h_{cr}$: stretching factor (nondimensional). 
    214200  The larger \np{ppacr}{ppacr}, the smaller the stretching. 
    215201  Values from $3$ to $10$ are usual. 
    216 \item 
    217   \np{ppkth}{ppkth}~$= h_{th}$: is approximately the model level at which maximum stretching occurs 
     202\item \np{ppkth}{ppkth}~$= h_{th}$: is approximately the model level at which maximum stretching occurs 
    218203  (nondimensional, usually of order 1/2 or 2/3 of \jp{jpk}) 
    219 \item 
    220   \np{ppdzmin}{ppdzmin}: minimum thickness for the top layer (in meters). 
    221 \item 
    222   \np{pphmax}{pphmax}: total depth of the ocean (meters). 
     204\item \np{ppdzmin}{ppdzmin}: minimum thickness for the top layer (in meters). 
     205\item \np{pphmax}{pphmax}: total depth of the ocean (meters). 
    223206\end{itemize} 
    224207 
     
    227210\np{pphmax}{pphmax}~$= 5750~m$. 
    228211 
    229 %% %>>>>>>>>>>>>>>>>>>>>>>>>>>>> 
    230212\begin{table} 
    231213  \centering 
     
    304286\end{table} 
    305287%%%YY 
    306 %>>>>>>>>>>>>>>>>>>>>>>>>>>>> 
    307288%% % ------------------------------------------------------------------------------------------------------------- 
    308289%% %        Meter Bathymetry 
     
    314295\np{nn_bathy}{nn\_bathy} (found in \nam{dom}{dom} namelist (\texttt{DOMAINCFG} variant) ): 
    315296\begin{description} 
    316 \item[{\np[=0]{nn_bathy}{nn\_bathy}}]: 
     297\item [{\np[=0]{nn_bathy}{nn\_bathy}}]: 
    317298  a flat-bottom domain is defined. 
    318299  The total depth $z_w (jpk)$ is given by the coordinate transformation. 
    319300  The domain can either be a closed basin or a periodic channel depending on the parameter \np{jperio}{jperio}. 
    320 \item[{\np[=-1]{nn_bathy}{nn\_bathy}}]: 
     301\item [{\np[=-1]{nn_bathy}{nn\_bathy}}]: 
    321302  a domain with a bump of topography one third of the domain width at the central latitude. 
    322303  This is meant for the "EEL-R5" configuration, a periodic or open boundary channel with a seamount. 
    323 \item[{\np[=1]{nn_bathy}{nn\_bathy}}]: 
     304\item [{\np[=1]{nn_bathy}{nn\_bathy}}]: 
    324305  read a bathymetry and ice shelf draft (if needed). 
    325306  The \ifile{bathy\_meter} file (Netcdf format) provides the ocean depth (positive, in meters) at 
     
    341322After reading the bathymetry, the algorithm for vertical grid definition differs between the different options: 
    342323\begin{description} 
    343 \item[\forcode{ln_zco = .true.}] 
     324\item [\forcode{ln_zco = .true.}] 
    344325  set a reference coordinate transformation $z_0(k)$, and set $z(i,j,k,t) = z_0(k)$ where $z_0(k)$ is the closest match to the depth at $(i,j)$. 
    345 \item[\forcode{ln_zps = .true.}] 
     326\item [\forcode{ln_zps = .true.}] 
    346327  set a reference coordinate transformation $z_0(k)$, and calculate the thickness of the deepest level at 
    347328  each $(i,j)$ point using the bathymetry, to obtain the final three-dimensional depth and scale factor arrays. 
    348 \item[\forcode{ln_sco = .true.}] 
     329\item [\forcode{ln_sco = .true.}] 
    349330  smooth the bathymetry to fulfill the hydrostatic consistency criteria and 
    350331  set the three-dimensional transformation. 
    351 \item[\forcode{s-z and s-zps}] 
     332\item [\forcode{s-z and s-zps}] 
    352333  smooth the bathymetry to fulfill the hydrostatic consistency criteria and 
    353334  set the three-dimensional transformation $z(i,j,k)$, 
     
    356337%%% 
    357338 
    358 % ------------------------------------------------------------------------------------------------------------- 
    359 %        z-coordinate with constant thickness 
    360 % ------------------------------------------------------------------------------------------------------------- 
    361339\subsubsection[$Z$-coordinate with uniform thickness levels (\forcode{ln_zco})]{$Z$-coordinate with uniform thickness levels (\protect\np{ln_zco}{ln\_zco})} 
    362340\label{subsec:DOMCFG_zco} 
     
    368346rarely used in modern simulations but it can be useful for testing purposes. 
    369347 
    370 % ------------------------------------------------------------------------------------------------------------- 
    371 %        z-coordinate with partial step 
    372 % ------------------------------------------------------------------------------------------------------------- 
    373348\subsubsection[$Z$-coordinate with partial step (\forcode{ln_zps})]{$Z$-coordinate with partial step (\protect\np{ln_zps}{ln\_zps})} 
    374349\label{subsec:DOMCFG_zps} 
     
    398373the default thickness $e_{3t}(jk)$). 
    399374 
    400 % ------------------------------------------------------------------------------------------------------------- 
    401 %        s-coordinate 
    402 % ------------------------------------------------------------------------------------------------------------- 
    403375\subsubsection[$S$-coordinate (\forcode{ln_sco})]{$S$-coordinate (\protect\np{ln_sco}{ln\_sco})} 
    404376\label{sec:DOMCFG_sco} 
     
    465437\] 
    466438 
    467 %% %>>>>>>>>>>>>>>>>>>>>>>>>>>>> 
    468439\begin{figure}[!ht] 
    469440  \centering 
     
    474445  \label{fig:DOMCFG_sco_function} 
    475446\end{figure} 
    476 %% %>>>>>>>>>>>>>>>>>>>>>>>>>>>> 
    477447 
    478448where $H_c$ is the critical depth (\np{rn_hc}{rn\_hc}) at which the coordinate transitions from pure $\sigma$ to 
     
    516486where the namelist parameters \np{rn_zb_a}{rn\_zb\_a} and \np{rn_zb_b}{rn\_zb\_b} are $a$ and $b$ respectively. 
    517487 
    518 %% %>>>>>>>>>>>>>>>>>>>>>>>>>>>> 
    519488\begin{figure}[!ht] 
    520489  \centering 
     
    550519and is output as part of the model mesh file at the start of the run. 
    551520 
    552 % ------------------------------------------------------------------------------------------------------------- 
    553 %        z*- or s*-coordinate 
    554 % ------------------------------------------------------------------------------------------------------------- 
    555521\subsubsection[\zstar- or \sstar-coordinate (\forcode{ln_linssh})]{\zstar- or \sstar-coordinate (\protect\np{ln_linssh}{ln\_linssh})} 
    556522\label{subsec:DOMCFG_zgr_star} 
  • NEMO/trunk/doc/latex/NEMO/subfiles/apdx_algos.tex

    r11584 r11596  
    22 
    33\begin{document} 
    4 % ================================================================ 
    5 % Appendix E : Note on some algorithms 
    6 % ================================================================ 
    74\chapter{Note on some algorithms} 
    85\label{apdx:ALGOS} 
     
    107\chaptertoc 
    118 
    12 \newpage 
    13  
    149This appendix some on going consideration on algorithms used or planned to be used in \NEMO. 
    1510 
    16 % ------------------------------------------------------------------------------------------------------------- 
    17 %        UBS scheme 
    18 % ------------------------------------------------------------------------------------------------------------- 
    1911\section{Upstream Biased Scheme (UBS) (\protect\np[=.true.]{ln_traadv_ubs}{ln\_traadv\_ubs})} 
    2012\label{sec:ALGOS_tra_adv_ubs} 
     
    188180which leads to ${A_u^{lT}} = \frac{1}{12} {e_{1u}}^3\ |u|$ 
    189181 
    190 % ------------------------------------------------------------------------------------------------------------- 
    191 %        Leap-Frog energetic 
    192 % ------------------------------------------------------------------------------------------------------------- 
    193182\section{Leapfrog energetic} 
    194183\label{sec:ALGOS_LF} 
     
    244233In time this boundary condition is not physical and \textbf{add something here!!!} 
    245234 
    246 % ================================================================ 
    247 % Iso-neutral diffusion : 
    248 % ================================================================ 
    249  
    250235\section{Lateral diffusion operator} 
    251236 
    252 % ================================================================ 
    253 % Griffies' iso-neutral diffusion operator : 
    254 % ================================================================ 
    255237\subsection{Griffies iso-neutral diffusion operator} 
    256238 
     
    305287the presence of partial cell at the ocean bottom (see \autoref{subsec:ALGOS_Gf_operator}). 
    306288 
    307 %>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> 
    308289\begin{figure}[!ht] 
    309290  \centering 
     
    315296  \label{fig:ALGOS_ISO_triad} 
    316297\end{figure} 
    317 %>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> 
    318298 
    319299The four iso-neutral fluxes associated with the triads are defined at $T$-point. 
     
    372352This expression of the iso-neutral diffusion has been chosen in order to satisfy the following six properties: 
    373353\begin{description} 
    374 \item[$\bullet$ horizontal diffusion] 
     354\item [$\bullet$ horizontal diffusion] 
    375355  The discretization of the diffusion operator recovers the traditional five-point Laplacian in 
    376356  the limit of flat iso-neutral direction: 
     
    383363  \] 
    384364 
    385 \item[$\bullet$ implicit treatment in the vertical] 
     365\item [$\bullet$ implicit treatment in the vertical] 
    386366  In the diagonal term associated with the vertical divergence of the iso-neutral fluxes 
    387367  \ie\ the term associated with a second order vertical derivative) 
     
    397377  can be quite large. 
    398378 
    399 \item[$\bullet$ pure iso-neutral operator] 
     379\item [$\bullet$ pure iso-neutral operator] 
    400380  The iso-neutral flux of locally referenced potential density is zero, \ie 
    401381  \begin{align*} 
     
    413393  the definition of the triads' slopes \autoref{eq:ALGOS_Gf_slopes}. 
    414394 
    415 \item[$\bullet$ conservation of tracer] 
     395\item [$\bullet$ conservation of tracer] 
    416396  The iso-neutral diffusion term conserve the total tracer content, \ie 
    417397  \[ 
     
    421401This property is trivially satisfied since the iso-neutral diffusive operator is written in flux form. 
    422402 
    423 \item[$\bullet$ decrease of tracer variance] 
     403\item [$\bullet$ decrease of tracer variance] 
    424404  The iso-neutral diffusion term does not increase the total tracer variance, \ie 
    425405  \[ 
     
    434414the field on which it is applied become free of grid-point noise. 
    435415 
    436 \item[$\bullet$ self-adjoint operator] 
     416\item [$\bullet$ self-adjoint operator] 
    437417  The iso-neutral diffusion operator is self-adjoint, \ie 
    438418  \[ 
     
    446426\end{description} 
    447427 
    448 % ================================================================ 
    449 % Skew flux formulation for Eddy Induced Velocity : 
    450 % ================================================================ 
    451428\subsection{Eddy induced velocity and skew flux formulation} 
    452429 
     
    602579 
    603580$\ $\newpage      %force an empty line 
    604 % ================================================================ 
    605 % Discrete Invariants of the iso-neutral diffrusion 
    606 % ================================================================ 
    607581\subsection{Discrete invariants of the iso-neutral diffrusion} 
    608582\label{subsec:ALGOS_Gf_operator} 
     
    767741There is no need to develop a specific to obtain it. 
    768742 
    769 \newpage 
    770  
    771 % ================================================================ 
    772 % Discrete Invariants of the skew flux formulation 
    773 % ================================================================ 
    774743\subsection{Discrete invariants of the skew flux formulation} 
    775744\label{subsec:ALGOS_eiv_skew} 
  • NEMO/trunk/doc/latex/NEMO/subfiles/apdx_diff_opers.tex

    r11584 r11596  
    22 
    33\begin{document} 
    4 % ================================================================ 
    5 % Chapter Appendix B : Diffusive Operators 
    6 % ================================================================ 
    74\chapter{Diffusive Operators} 
    85\label{apdx:DIFFOPERS} 
     
    107\chaptertoc 
    118 
    12 \newpage 
    13  
    14 % ================================================================ 
    15 % Horizontal/Vertical 2nd Order Tracer Diffusive Operators 
    16 % ================================================================ 
    179\section{Horizontal/Vertical $2^{nd}$ order tracer diffusive operators} 
    1810\label{sec:DIFFOPERS_1} 
     
    156148%\addtocounter{equation}{-2} 
    157149 
    158 % ================================================================ 
    159 % Isopycnal/Vertical 2nd Order Tracer Diffusive Operators 
    160 % ================================================================ 
    161150\section{Iso/Diapycnal $2^{nd}$ order tracer diffusive operators} 
    162151\label{sec:DIFFOPERS_2} 
     
    320309\autoref{sec:DIFFOPERS_1} onto $s$-coordinates is exact, however steep the $s$-surfaces. 
    321310 
    322  
    323 % ================================================================ 
    324 % Lateral/Vertical Momentum Diffusive Operators 
    325 % ================================================================ 
    326311\section{Lateral/Vertical momentum diffusive operators} 
    327312\label{sec:DIFFOPERS_3} 
  • NEMO/trunk/doc/latex/NEMO/subfiles/apdx_invariants.tex

    r11584 r11596  
    22 
    33\begin{document} 
    4 % ================================================================ 
    5 % Chapter Ñ Appendix C : Discrete Invariants of the Equations 
    6 % ================================================================ 
    74\chapter{Discrete Invariants of the Equations} 
    85\label{apdx:INVARIANTS} 
     
    1512%\gmcomment{ 
    1613 
    17 \newpage 
    18  
    19 % ================================================================ 
    20 % Introduction / Notations 
    21 % ================================================================ 
    2214\section{Introduction / Notations} 
    2315\label{sec:INVARIANTS_0} 
     
    9385\end{flalign} 
    9486 
    95 % ================================================================ 
    96 % Continuous Total energy Conservation 
    97 % ================================================================ 
    9887\section{Continuous conservation} 
    9988\label{sec:INVARIANTS_1} 
     
    322311% 
    323312 
    324 % ================================================================ 
    325 % Discrete Total energy Conservation : vector invariant form 
    326 % ================================================================ 
    327313\section{Discrete total energy conservation: vector invariant form} 
    328314\label{sec:INVARIANTS_2} 
    329315 
    330 % ------------------------------------------------------------------------------------------------------------- 
    331 %       Total energy conservation 
    332 % ------------------------------------------------------------------------------------------------------------- 
    333316\subsection{Total energy conservation} 
    334317\label{subsec:INVARIANTS_KE+PE_vect} 
     
    354337leads to the discrete equivalent of the four equations \autoref{eq:INVARIANTS_E_tot_flux}. 
    355338 
    356 % ------------------------------------------------------------------------------------------------------------- 
    357 %       Vorticity term (coriolis + vorticity part of the advection) 
    358 % ------------------------------------------------------------------------------------------------------------- 
    359339\subsection{Vorticity term (coriolis + vorticity part of the advection)} 
    360340\label{subsec:INVARIANTS_vor} 
     
    363343or the planetary ($q=f/e_{3f}$), or the total potential vorticity ($q=(\zeta +f) /e_{3f}$). 
    364344Two discretisation of the vorticity term (ENE and EEN) allows the conservation of the kinetic energy. 
    365 % ------------------------------------------------------------------------------------------------------------- 
    366 %       Vorticity Term with ENE scheme 
    367 % ------------------------------------------------------------------------------------------------------------- 
    368345\subsubsection{Vorticity term with ENE scheme (\protect\np[=.true.]{ln_dynvor_ene}{ln\_dynvor\_ene})} 
    369346\label{subsec:INVARIANTS_vorENE} 
     
    403380In other words, the domain averaged kinetic energy does not change due to the vorticity term. 
    404381 
    405 % ------------------------------------------------------------------------------------------------------------- 
    406 %       Vorticity Term with EEN scheme 
    407 % ------------------------------------------------------------------------------------------------------------- 
    408382\subsubsection{Vorticity term with EEN scheme (\protect\np[=.true.]{ln_dynvor_een}{ln\_dynvor\_een})} 
    409383\label{subsec:INVARIANTS_vorEEN_vect} 
     
    475449\end{flalign*} 
    476450 
    477 % ------------------------------------------------------------------------------------------------------------- 
    478 %       Gradient of Kinetic Energy / Vertical Advection 
    479 % ------------------------------------------------------------------------------------------------------------- 
    480451\subsubsection{Gradient of kinetic energy / Vertical advection} 
    481452\label{subsec:INVARIANTS_zad} 
     
    585556Blah blah required on the the step representation of bottom topography..... 
    586557 
    587  
    588 % ------------------------------------------------------------------------------------------------------------- 
    589 %       Pressure Gradient Term 
    590 % ------------------------------------------------------------------------------------------------------------- 
    591558\subsection{Pressure gradient term} 
    592559\label{subsec:INVARIANTS_2.6} 
     
    731698Nevertheless, it is almost never satisfied since a linear equation of state is rarely used. 
    732699 
    733 % ================================================================ 
    734 % Discrete Total energy Conservation : flux form 
    735 % ================================================================ 
    736700\section{Discrete total energy conservation: flux form} 
    737701\label{sec:INVARIANTS_3} 
    738702 
    739 % ------------------------------------------------------------------------------------------------------------- 
    740 %       Total energy conservation 
    741 % ------------------------------------------------------------------------------------------------------------- 
    742703\subsection{Total energy conservation} 
    743704\label{subsec:INVARIANTS_KE+PE_flux} 
     
    760721vector invariant or in flux form, leads to the discrete equivalent of the ???? 
    761722 
    762  
    763 % ------------------------------------------------------------------------------------------------------------- 
    764 %       Coriolis and advection terms: flux form 
    765 % ------------------------------------------------------------------------------------------------------------- 
    766723\subsection{Coriolis and advection terms: flux form} 
    767724\label{subsec:INVARIANTS_3.2} 
    768725 
    769 % ------------------------------------------------------------------------------------------------------------- 
    770 %       Coriolis plus ``metric'' Term 
    771 % ------------------------------------------------------------------------------------------------------------- 
    772726\subsubsection{Coriolis plus ``metric'' term} 
    773727\label{subsec:INVARIANTS_3.3} 
     
    788742The derivation is the same as for the vorticity term in the vector invariant form (\autoref{subsec:INVARIANTS_vor}). 
    789743 
    790 % ------------------------------------------------------------------------------------------------------------- 
    791 %       Flux form advection 
    792 % ------------------------------------------------------------------------------------------------------------- 
    793744\subsubsection{Flux form advection} 
    794745\label{subsec:INVARIANTS_3.4} 
     
    869820The horizontal kinetic energy is not conserved, but forced to decay (\ie\ the scheme is diffusive). 
    870821 
    871 % ================================================================ 
    872 % Discrete Enstrophy Conservation 
    873 % ================================================================ 
    874822\section{Discrete enstrophy conservation} 
    875823\label{sec:INVARIANTS_4} 
    876824 
    877 % ------------------------------------------------------------------------------------------------------------- 
    878 %       Vorticity Term with ENS scheme 
    879 % ------------------------------------------------------------------------------------------------------------- 
    880825\subsubsection{Vorticity term with ENS scheme  (\protect\np[=.true.]{ln_dynvor_ens}{ln\_dynvor\_ens})} 
    881826\label{subsec:INVARIANTS_vorENS} 
     
    944889The later equality is obtain only when the flow is horizontally non-divergent, \ie\ $\chi$=$0$. 
    945890 
    946 % ------------------------------------------------------------------------------------------------------------- 
    947 %       Vorticity Term with EEN scheme 
    948 % ------------------------------------------------------------------------------------------------------------- 
    949891\subsubsection{Vorticity Term with EEN scheme (\protect\np[=.true.]{ln_dynvor_een}{ln\_dynvor\_een})} 
    950892\label{subsec:INVARIANTS_vorEEN} 
     
    1017959\end{flalign*} 
    1018960 
    1019 % ================================================================ 
    1020 % Conservation Properties on Tracers 
    1021 % ================================================================ 
    1022961\section{Conservation properties on tracers} 
    1023962\label{sec:INVARIANTS_5} 
     
    1033972as the equation of state is non linear with respect to $T$ and $S$. 
    1034973In practice, the mass is conserved to a very high accuracy. 
    1035 % ------------------------------------------------------------------------------------------------------------- 
    1036 %       Advection Term 
    1037 % ------------------------------------------------------------------------------------------------------------- 
    1038974\subsection{Advection term} 
    1039975\label{subsec:INVARIANTS_5.1} 
     
    10991035which is the discrete form of $ \frac{1}{2} \int_D {  T^2 \frac{1}{e_3} \frac{\partial  e_3 }{\partial t} \;dv }$. 
    11001036 
    1101 % ================================================================ 
    1102 % Conservation Properties on Lateral Momentum Physics 
    1103 % ================================================================ 
    11041037\section{Conservation properties on lateral momentum physics} 
    11051038\label{sec:INVARIANTS_dynldf_properties} 
     
    11201053the term associated with the horizontal gradient of the divergence is locally zero. 
    11211054 
    1122 % ------------------------------------------------------------------------------------------------------------- 
    1123 %       Conservation of Potential Vorticity 
    1124 % ------------------------------------------------------------------------------------------------------------- 
    11251055\subsection{Conservation of potential vorticity} 
    11261056\label{subsec:INVARIANTS_6.1} 
     
    11541084\end{flalign*} 
    11551085 
    1156 % ------------------------------------------------------------------------------------------------------------- 
    1157 %       Dissipation of Horizontal Kinetic Energy 
    1158 % ------------------------------------------------------------------------------------------------------------- 
    11591086\subsection{Dissipation of horizontal kinetic energy} 
    11601087\label{subsec:INVARIANTS_6.2} 
     
    12061133\] 
    12071134 
    1208 % ------------------------------------------------------------------------------------------------------------- 
    1209 %       Dissipation of Enstrophy 
    1210 % ------------------------------------------------------------------------------------------------------------- 
    12111135\subsection{Dissipation of enstrophy} 
    12121136\label{subsec:INVARIANTS_6.3} 
     
    12301154\end{flalign*} 
    12311155 
    1232 % ------------------------------------------------------------------------------------------------------------- 
    1233 %       Conservation of Horizontal Divergence 
    1234 % ------------------------------------------------------------------------------------------------------------- 
    12351156\subsection{Conservation of horizontal divergence} 
    12361157\label{subsec:INVARIANTS_6.4} 
     
    12571178\end{flalign*} 
    12581179 
    1259 % ------------------------------------------------------------------------------------------------------------- 
    1260 %       Dissipation of Horizontal Divergence Variance 
    1261 % ------------------------------------------------------------------------------------------------------------- 
    12621180\subsection{Dissipation of horizontal divergence variance} 
    12631181\label{subsec:INVARIANTS_6.5} 
     
    12831201\end{flalign*} 
    12841202 
    1285 % ================================================================ 
    1286 % Conservation Properties on Vertical Momentum Physics 
    1287 % ================================================================ 
    12881203\section{Conservation properties on vertical momentum physics} 
    12891204\label{sec:INVARIANTS_7} 
     
    14541369\end{flalign*} 
    14551370 
    1456 % ================================================================ 
    1457 % Conservation Properties on Tracer Physics 
    1458 % ================================================================ 
    14591371\section{Conservation properties on tracer physics} 
    14601372\label{sec:INVARIANTS_8} 
     
    14661378As for the advection term, there is conservation of mass only if the Equation Of Seawater is linear. 
    14671379 
    1468 % ------------------------------------------------------------------------------------------------------------- 
    1469 %       Conservation of Tracers 
    1470 % ------------------------------------------------------------------------------------------------------------- 
    14711380\subsection{Conservation of tracers} 
    14721381\label{subsec:INVARIANTS_8.1} 
     
    14991408In fact, this property simply results from the flux form of the operator. 
    15001409 
    1501 % ------------------------------------------------------------------------------------------------------------- 
    1502 %       Dissipation of Tracer Variance 
    1503 % ------------------------------------------------------------------------------------------------------------- 
    15041410\subsection{Dissipation of tracer variance} 
    15051411\label{subsec:INVARIANTS_8.2} 
  • NEMO/trunk/doc/latex/NEMO/subfiles/apdx_s_coord.tex

    r11584 r11596  
    33\begin{document} 
    44 
    5 % ================================================================ 
    6 % Chapter Appendix A : Curvilinear s-Coordinate Equations 
    7 % ================================================================ 
    85\chapter{Curvilinear $s-$Coordinate Equations} 
    96\label{apdx:SCOORD} 
     
    2118\end{figure} 
    2219 
    23  
    24 \newpage 
    25  
    26 % ================================================================ 
    27 % Chain rule 
    28 % ================================================================ 
    2920\section{Chain rule for $s-$coordinates} 
    3021\label{sec:SCOORD_chain} 
     
    121112\end{equation} 
    122113 
    123  
    124 % ================================================================ 
    125 % continuity equation 
    126 % ================================================================ 
    127114\section{Continuity equation in $s-$coordinates} 
    128115\label{sec:SCOORD_continuity} 
     
    240227the contribution of the time variation of the vertical coordinate to the volume budget. 
    241228 
    242  
    243 % ================================================================ 
    244 % momentum equation 
    245 % ================================================================ 
    246229\section{Momentum equation in $s-$coordinate} 
    247230\label{sec:SCOORD_momentum} 
     
    571554\ie\ the volume flux across the moving $s$-surfaces per unit horizontal area. 
    572555 
    573  
    574 % ================================================================ 
    575 % Tracer equation 
    576 % ================================================================ 
    577556\section{Tracer equation} 
    578557\label{sec:SCOORD_tracer} 
  • NEMO/trunk/doc/latex/NEMO/subfiles/apdx_triads.tex

    r11584 r11596  
    1212 
    1313\begin{document} 
    14 % ================================================================ 
    15 % Iso-neutral diffusion : 
    16 % ================================================================ 
    1714\chapter{Iso-Neutral Diffusion and Eddy Advection using Triads} 
    1815\label{apdx:TRIADS} 
    1916 
    2017\chaptertoc 
    21  
    22 \newpage 
    2318 
    2419\section[Choice of \forcode{namtra\_ldf} namelist parameters]{Choice of \protect\nam{tra_ldf}{tra\_ldf} namelist parameters} 
     
    4237The options specific to the Griffies scheme include: 
    4338\begin{description} 
    44 \item[{\np{ln_triad_iso}{ln\_triad\_iso}}] 
     39\item [{\np{ln_triad_iso}{ln\_triad\_iso}}] 
    4540  See \autoref{sec:TRIADS_taper}. 
    4641  If this is set false (the default), 
     
    5348  giving an almost pure horizontal diffusive tracer flux within the mixed layer. 
    5449  This is similar to the tapering suggested by \citet{gerdes.koberle.ea_CD91}. See \autoref{subsec:TRIADS_Gerdes-taper} 
    55 \item[{\np{ln_botmix_triad}{ln\_botmix\_triad}}] 
     50\item [{\np{ln_botmix_triad}{ln\_botmix\_triad}}] 
    5651  See \autoref{sec:TRIADS_iso_bdry}. 
    5752  If this is set false (the default) then the lateral diffusive fluxes 
     
    5954  If it is set true, however, then these lateral diffusive fluxes are applied, 
    6055  giving smoother bottom tracer fields at the cost of introducing diapycnal mixing. 
    61 \item[{\np{rn_sw_triad}{rn\_sw\_triad}}] 
     56\item [{\np{rn_sw_triad}{rn\_sw\_triad}}] 
    6257  blah blah to be added.... 
    6358\end{description} 
    6459The options shared with the Standard scheme include: 
    6560\begin{description} 
    66 \item[{\np{ln_traldf_msc}{ln\_traldf\_msc}}]   blah blah to be added 
    67 \item[{\np{rn_slpmax}{rn\_slpmax}}]  blah blah to be added 
     61\item [{\np{ln_traldf_msc}{ln\_traldf\_msc}}]   blah blah to be added 
     62\item [{\np{rn_slpmax}{rn\_slpmax}}]  blah blah to be added 
    6863\end{description} 
    6964 
     
    548543The diffusion scheme satisfies the following six properties: 
    549544\begin{description} 
    550 \item[$\bullet$ horizontal diffusion] 
     545\item [$\bullet$ horizontal diffusion] 
    551546  The discretization of the diffusion operator recovers the traditional five-point Laplacian 
    552547  \autoref{eq:TRIADS_lat-normal} in the limit of flat iso-neutral direction: 
     
    559554  \] 
    560555 
    561 \item[$\bullet$ implicit treatment in the vertical] 
     556\item [$\bullet$ implicit treatment in the vertical] 
    562557  Only tracer values associated with a single water column appear in the expression \autoref{eq:TRIADS_i33} for 
    563558  the $_{33}$ fluxes, vertical fluxes driven by vertical gradients. 
     
    575570  (where $b_w= e_{1w}\,e_{2w}\,e_{3w}$ is the volume of $w$-cells) can be quite large. 
    576571 
    577 \item[$\bullet$ pure iso-neutral operator] 
     572\item [$\bullet$ pure iso-neutral operator] 
    578573  The iso-neutral flux of locally referenced potential density is zero. 
    579574  See \autoref{eq:TRIADS_latflux-rho} and \autoref{eq:TRIADS_vertflux-triad2}. 
    580575 
    581 \item[$\bullet$ conservation of tracer] 
     576\item [$\bullet$ conservation of tracer] 
    582577  The iso-neutral diffusion conserves tracer content, \ie 
    583578  \[ 
     
    587582  This property is trivially satisfied since the iso-neutral diffusive operator is written in flux form. 
    588583 
    589 \item[$\bullet$ no increase of tracer variance] 
     584\item [$\bullet$ no increase of tracer variance] 
    590585  The iso-neutral diffusion does not increase the tracer variance, \ie 
    591586  \[ 
     
    600595  the field on which it is applied becomes free of grid-point noise. 
    601596 
    602 \item[$\bullet$ self-adjoint operator] 
     597\item [$\bullet$ self-adjoint operator] 
    603598  The iso-neutral diffusion operator is self-adjoint, \ie 
    604599  \begin{equation} 
     
    753748described above by \autoref{eq:TRIADS_Rtilde}. 
    754749\begin{enumerate} 
    755 \item 
    756   Mixed-layer depth is defined so as to avoid including regions of weak vertical stratification in 
     750\item Mixed-layer depth is defined so as to avoid including regions of weak vertical stratification in 
    757751  the slope definition. 
    758752  At each $i,j$ (simplified to $i$ in \autoref{fig:TRIADS_MLB_triad}), 
     
    766760  output the diagnosed mixed-layer depth $h_{\mathrm{ML}}=|z_{W}|_{k_{\mathrm{ML}}+1/2}$, 
    767761  the depth of the $w$-point above the $i,k_{\mathrm{ML}}$ tracer point. 
    768 \item 
    769   We define `basal' triad slopes ${\:}_i{\mathbb{R}_{\mathrm{base}}}_{\,i_p}^{k_p}$ as 
     762\item We define `basal' triad slopes ${\:}_i{\mathbb{R}_{\mathrm{base}}}_{\,i_p}^{k_p}$ as 
    770763  the slopes of those triads whose vertical `arms' go down from the $i,k_{\mathrm{ML}}$ tracer point to 
    771764  the $i,k_{\mathrm{ML}}-1$ tracer point below. 
     
    790783one gridbox deeper than the diagnosed ML depth $z_{\mathrm{ML}})$ that sets the $h$ used to taper the slopes in 
    791784\autoref{eq:TRIADS_rmtilde}. 
    792 \item 
    793   Finally, we calculate the adjusted triads ${\:}_i^k{\mathbb{R}_{\mathrm{ML}}}_{\,i_p}^{k_p}$ within 
     785\item Finally, we calculate the adjusted triads ${\:}_i^k{\mathbb{R}_{\mathrm{ML}}}_{\,i_p}^{k_p}$ within 
    794786  the mixed layer, by multiplying the appropriate ${\:}_i{\mathbb{R}_{\mathrm{base}}}_{\,i_p}^{k_p}$ by 
    795787  the ratio of the depth of the $w$-point ${z_w}_{k+k_p}$ to ${z_{\mathrm{base}}}_{\,i}$. 
     
    872864% This may give strange looking results, 
    873865% particularly where the mixed-layer depth varies strongly laterally. 
    874 % ================================================================ 
    875 % Skew flux formulation for Eddy Induced Velocity : 
    876 % ================================================================ 
    877866\section{Eddy induced advection formulated as a skew flux} 
    878867\label{sec:TRIADS_skew-flux} 
  • NEMO/trunk/doc/latex/NEMO/subfiles/chap_ASM.tex

    r11584 r11596  
    22 
    33\begin{document} 
    4 % ================================================================ 
    5 % Chapter Assimilation increments (ASM) 
    6 % ================================================================ 
    74\chapter{Apply Assimilation Increments (ASM)} 
    85\label{chap:ASM} 
     
    1916\end{tabular} 
    2017\end{figure} 
    21  
    22 \newpage 
    2318 
    2419The ASM code adds the functionality to apply increments to the model variables: temperature, salinity, 
     
    138133This specifies the number of iterations of the divergence damping. Setting a value of the order of 100 will result in a significant reduction in the vertical velocity induced by the increments. 
    139134 
    140  
    141135%========================================================================== 
    142136 
  • NEMO/trunk/doc/latex/NEMO/subfiles/chap_DIA.tex

    r11584 r11596  
    22 
    33\begin{document} 
    4 % ================================================================ 
    5 % Chapter I/O & Diagnostics 
    6 % ================================================================ 
    74\chapter{Output and Diagnostics (IOM, DIA, TRD, FLO)} 
    85\label{chap:DIA} 
     
    2320\end{figure} 
    2421 
    25 \newpage 
    26  
    27 % ================================================================ 
    28 %       Old Model Output 
    29 % ================================================================ 
    3022\section{Model output} 
    3123\label{sec:DIA_io_old} 
     
    5345%\gmcomment{                    % start of gmcomment 
    5446 
    55 % ================================================================ 
    56 % Diagnostics 
    57 % ================================================================ 
    5847\section{Standard model output (IOM)} 
    5948\label{sec:DIA_iom} 
     
    6453 
    6554\begin{enumerate} 
    66 \item 
    67   The complete and flexible control of the output files through external XML files adapted by 
     55\item The complete and flexible control of the output files through external XML files adapted by 
    6856  the user from standard templates. 
    69 \item 
    70   To achieve high performance and scalable output through the optional distribution of 
     57\item To achieve high performance and scalable output through the optional distribution of 
    7158  all diagnostic output related tasks to dedicated processes. 
    7259\end{enumerate} 
     
    7663 
    7764\begin{itemize} 
    78 \item 
    79   The choice of output frequencies that can be different for each file (including real months and years). 
    80 \item 
    81   The choice of file contents; includes complete flexibility over which data are written in which files 
     65\item The choice of output frequencies that can be different for each file (including real months and years). 
     66\item The choice of file contents; includes complete flexibility over which data are written in which files 
    8267  (the same data can be written in different files). 
    83 \item 
    84   The possibility to split output files at a chosen frequency. 
    85 \item 
    86   The possibility to extract a vertical or an horizontal subdomain. 
    87 \item 
    88   The choice of the temporal operation to perform, \eg: average, accumulate, instantaneous, min, max and once. 
    89 \item 
    90   Control over metadata via a large XML "database" of possible output fields. 
     68\item The possibility to split output files at a chosen frequency. 
     69\item The possibility to extract a vertical or an horizontal subdomain. 
     70\item The choice of the temporal operation to perform, \eg: average, accumulate, instantaneous, min, max and once. 
     71\item Control over metadata via a large XML "database" of possible output fields. 
    9172\end{itemize} 
    9273 
     
    150131If an additional variable must be written to a restart file, the following steps are needed: 
    151132\begin{description} 
    152    \item[step 1:] add variable name to a list of restart variables (in subroutine \rou{iom\_set\_rst\_vars,} \mdl{iom}) and 
     133   \item [step 1:] add variable name to a list of restart variables (in subroutine \rou{iom\_set\_rst\_vars,} \mdl{iom}) and 
    153134define correct grid for the variable (\forcode{grid_N_3D} - 3D variable, \forcode{grid_N} - 2D variable, \forcode{grid_vector} - 
    1541351D variable, \forcode{grid_scalar} - scalar), 
    155    \item[step 2:] add variable to the list of fields written by restart.  This can be done either in subroutine 
     136   \item [step 2:] add variable to the list of fields written by restart.  This can be done either in subroutine 
    156137\rou{iom\_set\_rstw\_core} (\mdl{iom}) or by calling  \rou{iom\_set\_rstw\_active} (\mdl{iom}) with the name of a variable 
    157138as an argument. This convention follows approach for writing restart using iom, where variables are 
     
    159140\end{description} 
    160141 
    161  
    162142An older versions of XIOS do not support reading functionality. It's recommended to use at least XIOS2@1451. 
    163  
    164143 
    165144\subsection{XIOS: XML Inputs-Outputs Server} 
     
    276255 
    277256\begin{enumerate} 
    278 \item[1.] 
     257\item [1.] 
    279258  in \NEMO\ code, add a \forcode{CALL iom_put( 'identifier', array )} where you want to output a 2D or 3D array. 
    280 \item[2.] 
     259\item [2.] 
    281260  If necessary, add \forcode{USE iom ! I/O manager library} to the list of used modules in 
    282261  the upper part of your module. 
    283 \item[3.] 
     262\item [3.] 
    284263  in the field\_def.xml file, add the definition of your variable using the same identifier you used in the f90 code 
    285264  (see subsequent sections for a details of the XML syntax and rules). 
     
    310289\xmlcode{<field_group id="SBC" ...>} which has been defined with the correct frequency of operations 
    311290(iom\_set\_field\_attr in \mdl{iom}) 
    312 \item[4.] 
     291\item [4.] 
    313292  add your field in one of the output files defined in iodef.xml 
    314293  (again see subsequent sections for syntax and rules) 
     
    737716 
    738717\begin{enumerate} 
    739 \item 
    740   Simple computation: directly define the computation when refering to the variable in the file definition. 
     718\item Simple computation: directly define the computation when refering to the variable in the file definition. 
    741719 
    742720\begin{xmllines} 
     
    746724\end{xmllines} 
    747725 
    748 \item 
    749   Simple computation: define a new variable and use it in the file definition. 
     726\item Simple computation: define a new variable and use it in the file definition. 
    750727 
    751728in field\_definition: 
     
    764741sst2 won't be evaluated. 
    765742 
    766 \item 
    767   Change of variable precision: 
     743\item Change of variable precision: 
    768744 
    769745\begin{xmllines} 
     
    778754Forcing double precision outputs with prec="8" (for example in the field\_definition) will avoid this problem. 
    779755 
    780 \item 
    781   add user defined attributes: 
     756\item add user defined attributes: 
    782757 
    783758\begin{xmllines} 
     
    794769\end{xmllines} 
    795770 
    796 \item 
    797   use of the ``@'' function: example 1, weighted temporal average 
     771\item use of the ``@'' function: example 1, weighted temporal average 
    798772 
    799773 - define a new variable in field\_definition 
     
    823797Note that in this case, freq\_op must be equal to the file output\_freq. 
    824798 
    825 \item 
    826   use of the ``@'' function: example 2, monthly SSH standard deviation 
     799\item use of the ``@'' function: example 2, monthly SSH standard deviation 
    827800 
    828801 - define a new variable in field\_definition 
     
    854827Note that in this case, freq\_op must be equal to the file output\_freq. 
    855828 
    856 \item 
    857   use of the ``@'' function: example 3, monthly average of SST diurnal cycle 
     829\item use of the ``@'' function: example 3, monthly average of SST diurnal cycle 
    858830 
    859831 - define 2 new variables in field\_definition 
     
    13261298This must be set to true if these metadata are to be included in the output files. 
    13271299 
    1328  
    1329 % ================================================================ 
    1330 %       NetCDF4 support 
    1331 % ================================================================ 
    13321300\section[NetCDF4 support (\texttt{\textbf{key\_netcdf4}})]{NetCDF4 support (\protect\key{netcdf4})} 
    13331301\label{sec:DIA_nc4} 
     
    14461414the invidual processing regions and different chunking choices may be desired. 
    14471415 
    1448 % ------------------------------------------------------------------------------------------------------------- 
    1449 %       Tracer/Dynamics Trends 
    1450 % ------------------------------------------------------------------------------------------------------------- 
    14511416\section[Tracer/Dynamics trends (\forcode{&namtrd})]{Tracer/Dynamics trends (\protect\nam{trd}{trd})} 
    14521417\label{sec:DIA_trd} 
     
    14701435 
    14711436\begin{description} 
    1472 \item[{\np{ln_glo_trd}{ln\_glo\_trd}}]: 
     1437\item [{\np{ln_glo_trd}{ln\_glo\_trd}}]: 
    14731438  at each \np{nn_trd}{nn\_trd} time-step a check of the basin averaged properties of 
    14741439  the momentum and tracer equations is performed. 
    14751440  This also includes a check of $T^2$, $S^2$, $\tfrac{1}{2} (u^2+v2)$, 
    14761441  and potential energy time evolution equations properties; 
    1477 \item[{\np{ln_dyn_trd}{ln\_dyn\_trd}}]: 
     1442\item [{\np{ln_dyn_trd}{ln\_dyn\_trd}}]: 
    14781443  each 3D trend of the evolution of the two momentum components is output; 
    1479 \item[{\np{ln_dyn_mxl}{ln\_dyn\_mxl}}]: 
     1444\item [{\np{ln_dyn_mxl}{ln\_dyn\_mxl}}]: 
    14801445  each 3D trend of the evolution of the two momentum components averaged over the mixed layer is output; 
    1481 \item[{\np{ln_vor_trd}{ln\_vor\_trd}}]: 
     1446\item [{\np{ln_vor_trd}{ln\_vor\_trd}}]: 
    14821447  a vertical summation of the moment tendencies is performed, 
    14831448  then the curl is computed to obtain the barotropic vorticity tendencies which are output; 
    1484 \item[{\np{ln_KE_trd}{ln\_KE\_trd}}] : 
     1449\item [{\np{ln_KE_trd}{ln\_KE\_trd}}] : 
    14851450  each 3D trend of the Kinetic Energy equation is output; 
    1486 \item[{\np{ln_tra_trd}{ln\_tra\_trd}}]: 
     1451\item [{\np{ln_tra_trd}{ln\_tra\_trd}}]: 
    14871452  each 3D trend of the evolution of temperature and salinity is output; 
    1488 \item[{\np{ln_tra_mxl}{ln\_tra\_mxl}}]: 
     1453\item [{\np{ln_tra_mxl}{ln\_tra\_mxl}}]: 
    14891454  each 2D trend of the evolution of temperature and salinity averaged over the mixed layer is output; 
    14901455\end{description} 
     
    14971462and none of the options have been tested with variable volume (\ie\ \np[=.true.]{ln_linssh}{ln\_linssh}). 
    14981463 
    1499 % ------------------------------------------------------------------------------------------------------------- 
    1500 %       On-line Floats trajectories 
    1501 % ------------------------------------------------------------------------------------------------------------- 
    15021464\section[FLO: On-Line Floats trajectories (\texttt{\textbf{key\_floats}})]{FLO: On-Line Floats trajectories (\protect\key{floats})} 
    15031465\label{sec:DIA_FLO} 
     
    16021564\end{xmllines} 
    16031565 
    1604  
    1605 % ------------------------------------------------------------------------------------------------------------- 
    1606 %       Harmonic analysis of tidal constituents 
    1607 % ------------------------------------------------------------------------------------------------------------- 
    16081566\section[Harmonic analysis of tidal constituents (\texttt{\textbf{key\_diaharm}})]{Harmonic analysis of tidal constituents (\protect\key{diaharm})} 
    16091567\label{sec:DIA_diag_harm} 
     
    16541612We obtain in output $C_{j}$ and $S_{j}$ for each tidal wave. 
    16551613 
    1656 % ------------------------------------------------------------------------------------------------------------- 
    1657 %       Sections transports 
    1658 % ------------------------------------------------------------------------------------------------------------- 
    16591614\section[Transports across sections (\texttt{\textbf{key\_diadct}})]{Transports across sections (\protect\key{diadct})} 
    16601615\label{sec:DIA_diag_dct} 
     
    18001755\end{table} 
    18011756 
    1802 % ================================================================ 
    1803 % Steric effect in sea surface height 
    1804 % ================================================================ 
    18051757\section{Diagnosing the steric effect in sea surface height} 
    18061758\label{sec:DIA_steric} 
    1807  
    18081759 
    18091760Changes in steric sea level are caused when changes in the density of the water column imply an expansion or 
     
    19801931Both steric and thermosteric sea level are computed in \mdl{diaar5}. 
    19811932 
    1982 % ------------------------------------------------------------------------------------------------------------- 
    1983 %       Other Diagnostics 
    1984 % ------------------------------------------------------------------------------------------------------------- 
    19851933\section{Other diagnostics} 
    19861934\label{sec:DIA_diag_others} 
     
    20011949- the depth of the thermocline (maximum of the vertical temperature gradient) (\mdl{diahth}) 
    20021950 
    2003  
    2004 %>>>>>>>>>>>>>>>>>>>>>>>>>>>> 
    20051951\begin{figure}[!t] 
    20061952  \centering 
     
    20191965  \label{fig:DIA_mask_subasins} 
    20201966\end{figure} 
    2021 %>>>>>>>>>>>>>>>>>>>>>>>>>>>> 
    20221967 
    20231968% ----------------------------------------------------------- 
  • NEMO/trunk/doc/latex/NEMO/subfiles/chap_DIU.tex

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    22 
    33\begin{document} 
    4 % ================================================================ 
    5 % Diurnal SST models (DIU) 
    6 % Edited by James While 
    7 % ================================================================ 
    84\chapter{Diurnal SST Models (DIU)} 
    95\label{chap:DIU} 
     
    117\chaptertoc 
    128 
    13  
    14 \newpage 
    159$\ $\newline % force a new line 
    1610 
     
    1913The skin temperature can be split into three parts: 
    2014\begin{itemize} 
    21 \item 
    22   A foundation SST which is free from diurnal warming. 
    23 \item 
    24   A warm layer, typically ~3\,m thick, 
     15\item A foundation SST which is free from diurnal warming. 
     16\item A warm layer, typically ~3\,m thick, 
    2517  where heating from solar radiation can cause a warm stably stratified layer during the daytime 
    26 \item 
    27   A cool skin, a thin layer, approximately ~1\, mm thick, 
     18\item A cool skin, a thin layer, approximately ~1\, mm thick, 
    2819  where long wave cooling is dominant and cools the immediate ocean surface. 
    2920\end{itemize} 
     
    4637This namelist contains only two variables: 
    4738\begin{description} 
    48 \item[{\np{ln_diurnal}{ln\_diurnal}}] 
     39\item [{\np{ln_diurnal}{ln\_diurnal}}] 
    4940  A logical switch for turning on/off both the cool skin and warm layer. 
    50 \item[{\np{ln_diurnal_only}{ln\_diurnal\_only}}] 
     41\item [{\np{ln_diurnal_only}{ln\_diurnal\_only}}] 
    5142  A logical switch which if \forcode{.true.} will run the diurnal model without the other dynamical parts of \NEMO. 
    5243  \np{ln_diurnal_only}{ln\_diurnal\_only} must be \forcode{.false.} if \np{ln_diurnal}{ln\_diurnal} is \forcode{.false.}. 
  • NEMO/trunk/doc/latex/NEMO/subfiles/chap_DOM.tex

    r11584 r11596  
    22 
    33\begin{document} 
    4 % ================================================================ 
    5 % Chapter 2 ——— Space and Time Domain (DOM) 
    6 % ================================================================ 
    74\chapter{Space Domain (DOM)} 
    85\label{chap:DOM} 
     
    3633\end{table} 
    3734 
    38 \newpage 
    39  
    4035Having defined the continuous equations in \autoref{chap:MB} and chosen a time discretisation \autoref{chap:TD}, 
    4136we need to choose a grid for spatial discretisation and related numerical algorithms. 
     
    4338and other relevant information about the DOM (DOMain) source code modules. 
    4439 
    45 % ================================================================ 
    46 % Fundamentals of the Discretisation 
    47 % ================================================================ 
    4840\section{Fundamentals of the discretisation} 
    4941\label{sec:DOM_basics} 
    5042 
    51 % ------------------------------------------------------------------------------------------------------------- 
    52 %        Arrangement of Variables 
    53 % ------------------------------------------------------------------------------------------------------------- 
    5443\subsection{Arrangement of variables} 
    5544\label{subsec:DOM_cell} 
    5645 
    57 %>>>>>>>>>>>>>>>>>>>>>>>>>>>> 
    5846\begin{figure}[!tb] 
    5947  \centering 
     
    6856  \label{fig:DOM_cell} 
    6957\end{figure} 
    70 %>>>>>>>>>>>>>>>>>>>>>>>>>>>> 
    7158 
    7259The numerical techniques used to solve the Primitive Equations in this model are based on the traditional, 
     
    9986(see \autoref{eq:DOM_bar} in the next section). 
    10087 
    101 %>>>>>>>>>>>>>>>>>>>>>>>>>>>> 
    10288\begin{table}[!tb] 
    10389  \centering 
     
    130116  \label{tab:DOM_cell} 
    131117\end{table} 
    132 %>>>>>>>>>>>>>>>>>>>>>>>>>>>> 
    133118 
    134119Note that the definition of the scale factors 
     
    145130(rather than allowing the user to set arbitrary jumps in thickness between adjacent layers) \citep{treguier.dukowicz.ea_JGR96}. 
    146131An example of the effect of such a choice is shown in \autoref{fig:DOM_zgr_e3}. 
    147 %>>>>>>>>>>>>>>>>>>>>>>>>>>>> 
    148132\begin{figure}[!t] 
    149133  \centering 
     
    160144  \label{fig:DOM_zgr_e3} 
    161145\end{figure} 
    162 %>>>>>>>>>>>>>>>>>>>>>>>>>>>> 
    163  
    164 % ------------------------------------------------------------------------------------------------------------- 
    165 %        Vector Invariant Formulation 
    166 % ------------------------------------------------------------------------------------------------------------- 
     146 
    167147\subsection{Discrete operators} 
    168148\label{subsec:DOM_operators} 
     
    255235demonstrate integral conservative properties of the discrete formulation chosen. 
    256236 
    257 % ------------------------------------------------------------------------------------------------------------- 
    258 %        Numerical Indexing 
    259 % ------------------------------------------------------------------------------------------------------------- 
    260237\subsection{Numerical indexing} 
    261238\label{subsec:DOM_Num_Index} 
    262239 
    263 %>>>>>>>>>>>>>>>>>>>>>>>>>>>> 
    264240\begin{figure}[!tb] 
    265241  \centering 
     
    271247  \label{fig:DOM_index_hor} 
    272248\end{figure} 
    273 %>>>>>>>>>>>>>>>>>>>>>>>>>>>> 
    274249 
    275250The array representation used in the \fortran\ code requires an integer indexing. 
     
    315290accommodate the opposing vertical index directions in implementation and documentation. 
    316291 
    317 %>>>>>>>>>>>>>>>>>>>>>>>>>>>> 
    318292\begin{figure}[!pt] 
    319293  \centering 
     
    326300  \label{fig:DOM_index_vert} 
    327301\end{figure} 
    328 %>>>>>>>>>>>>>>>>>>>>>>>>>>>> 
    329  
    330 % ------------------------------------------------------------------------------------------------------------- 
    331 %        Domain configuration 
    332 % ------------------------------------------------------------------------------------------------------------- 
     302 
    333303\section{Spatial domain configuration} 
    334304\label{subsec:DOM_config} 
     
    385355See \autoref{sec:LBC_jperio} for details on the available options and the corresponding values for \jp{jperio}. 
    386356 
    387 % ================================================================ 
    388 % Domain: Horizontal Grid (mesh) 
    389 % ================================================================ 
    390357\subsection[Horizontal grid mesh (\textit{domhgr.F90}]{Horizontal grid mesh (\protect\mdl{domhgr})} 
    391358\label{subsec:DOM_hgr} 
    392359 
    393 % ================================================================ 
    394 % Domain: List of hgr-related fields needed 
    395 % ================================================================ 
    396360\subsubsection{Required fields} 
    397361\label{sec:DOM_hgr_fields} 
     
    453417thus no specific arrays are defined at $w$ points. 
    454418 
    455  
    456 % ================================================================ 
    457 % Domain: Vertical Grid (domzgr) 
    458 % ================================================================ 
    459419\subsection[Vertical grid (\textit{domzgr.F90})]{Vertical grid (\protect\mdl{domzgr})} 
    460420\label{subsec:DOM_zgr} 
     
    476436\end{enumerate} 
    477437 
    478 %>>>>>>>>>>>>>>>>>>>>>>>>>>>> 
    479438\begin{figure}[!tb] 
    480439  \centering 
     
    491450  \label{fig:DOM_z_zps_s_sps} 
    492451\end{figure} 
    493 %>>>>>>>>>>>>>>>>>>>>>>>>>>>> 
    494452 
    495453The choice of a vertical coordinate is made when setting up the configuration; 
     
    583541With ice cavities, \jp{top\_level} determines the first wet point below the overlying ice shelf. 
    584542 
    585  
    586 % ------------------------------------------------------------------------------------------------------------- 
    587 %        level bathymetry and mask 
    588 % ------------------------------------------------------------------------------------------------------------- 
    589543\subsubsection{Level bathymetry and mask} 
    590544\label{subsec:DOM_msk} 
    591  
    592545 
    593546From \jp{top\_level} and \jp{bottom\_level} fields, the mask fields are defined as follows: 
     
    619572%% (see \autoref{fig:LBC_jperio}). 
    620573 
    621  
    622574%------------------------------------------------------------------------------------------------- 
    623575%        Closed seas 
     
    643595\end{clines} 
    644596 
    645 % ------------------------------------------------------------------------------------------------------------- 
    646 %        Grid files 
    647 % ------------------------------------------------------------------------------------------------------------- 
    648597\subsection{Output grid files} 
    649598\label{subsec:DOM_meshmask} 
     
    664613This file contains additional fields that can be useful for post-processing applications. 
    665614 
    666 % ================================================================ 
    667 % Domain: Initial State (dtatsd & istate) 
    668 % ================================================================ 
    669615\section[Initial state (\textit{istate.F90} and \textit{dtatsd.F90})]{Initial state (\protect\mdl{istate} and \protect\mdl{dtatsd})} 
    670616\label{sec:DOM_DTA_tsd} 
     
    682628 
    683629\begin{description} 
    684 \item[{\np[=.true.]{ln_tsd_init}{ln\_tsd\_init}}] 
     630\item [{\np[=.true.]{ln_tsd_init}{ln\_tsd\_init}}] 
    685631  Use T and S input files that can be given on the model grid itself or on their native input data grids. 
    686632  In the latter case, the data will be interpolated on-the-fly both in the horizontal and the vertical to the model grid 
     
    688634  The information relating to the input files are specified in the \np{sn_tem}{sn\_tem} and \np{sn_sal}{sn\_sal} structures. 
    689635  The computation is done in the \mdl{dtatsd} module. 
    690 \item[{\np[=.false.]{ln_tsd_init}{ln\_tsd\_init}}] 
     636\item [{\np[=.false.]{ln_tsd_init}{ln\_tsd\_init}}] 
    691637  Initial values for T and S are set via a user supplied \rou{usr\_def\_istate} routine contained in \mdl{userdef\_istate}. 
    692638  The default version sets horizontally uniform T and profiles as used in the GYRE configuration 
  • NEMO/trunk/doc/latex/NEMO/subfiles/chap_DYN.tex

    r11584 r11596  
    22 
    33\begin{document} 
    4 % ================================================================ 
    5 % Chapter ——— Ocean Dynamics (DYN) 
    6 % ================================================================ 
    74\chapter{Ocean Dynamics (DYN)} 
    85\label{chap:DYN} 
     
    5653MISC correspond to "extracting tendency terms" or "vorticity balance"?} 
    5754 
    58 % ================================================================ 
    59 % Sea Surface Height evolution & Diagnostics variables 
    60 % ================================================================ 
    6155\section{Sea surface height and diagnostic variables ($\eta$, $\zeta$, $\chi$, $w$)} 
    6256\label{sec:DYN_divcur_wzv} 
     
    158152(see \autoref{subsec:DOM_Num_Index_vertical}). 
    159153 
    160  
    161 % ================================================================ 
    162 % Coriolis and Advection terms: vector invariant form 
    163 % ================================================================ 
    164154\section{Coriolis and advection: vector invariant form} 
    165155\label{sec:DYN_adv_cor_vect} 
     
    182172\autoref{chap:LBC}. 
    183173 
    184 % ------------------------------------------------------------------------------------------------------------- 
    185 %        Vorticity term 
    186 % ------------------------------------------------------------------------------------------------------------- 
    187174\subsection[Vorticity term (\textit{dynvor.F90})]{Vorticity term (\protect\mdl{dynvor})} 
    188175\label{subsec:DYN_vor} 
     
    314301\end{equation} 
    315302 
    316 %>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> 
    317303\begin{figure}[!ht] 
    318304  \centering 
     
    414400an option which is only available with a TVD scheme (see \np{ln_traadv_tvd_zts}{ln\_traadv\_tvd\_zts} in \autoref{subsec:TRA_adv_tvd}). 
    415401 
    416  
    417 % ================================================================ 
    418 % Coriolis and Advection : flux form 
    419 % ================================================================ 
    420402\section{Coriolis and advection: flux form} 
    421403\label{sec:DYN_adv_cor_flux} 
     
    430412At the lateral boundaries either free slip, 
    431413no slip or partial slip boundary conditions are applied following \autoref{chap:LBC}. 
    432  
    433414 
    434415%-------------------------------------------------------------------------------------------------------------- 
     
    562543%%% 
    563544 
    564 % ================================================================ 
    565 %           Hydrostatic pressure gradient term 
    566 % ================================================================ 
    567545\section[Hydrostatic pressure gradient (\textit{dynhpg.F90})]{Hydrostatic pressure gradient (\protect\mdl{dynhpg})} 
    568546\label{sec:DYN_hpg} 
     
    780758This option is controlled by  \np{nn_dynhpg_rst}{nn\_dynhpg\_rst}, a namelist parameter. 
    781759 
    782 % ================================================================ 
    783 % Surface Pressure Gradient 
    784 % ================================================================ 
    785760\section[Surface pressure gradient (\textit{dynspg.F90})]{Surface pressure gradient (\protect\mdl{dynspg})} 
    786761\label{sec:DYN_spg} 
     
    809784so that the update of the next velocities is done in module \mdl{dynspg\_flt} and not in \mdl{dynnxt}. 
    810785 
    811  
    812786The form of the surface pressure gradient term depends on how the user wants to 
    813787handle the fast external gravity waves that are a solution of the analytical equation (\autoref{sec:MB_hor_pg}). 
     
    820794The extra term introduced in the filtered method is calculated implicitly, so that a solver is used to compute it. 
    821795As a consequence the update of the $next$ velocities is done in module \mdl{dynspg\_flt} and not in \mdl{dynnxt}. 
    822  
    823796 
    824797%-------------------------------------------------------------------------------------------------------------- 
     
    10921065%>>>>>=============== 
    10931066 
    1094  
    10951067%-------------------------------------------------------------------------------------------------------------- 
    10961068% Filtered free surface formulation 
     
    11211093It is computed once and for all and applies to all ocean time steps. 
    11221094 
    1123 % ================================================================ 
    1124 % Lateral diffusion term 
    1125 % ================================================================ 
    11261095\section[Lateral diffusion term and operators (\textit{dynldf.F90})]{Lateral diffusion term and operators (\protect\mdl{dynldf})} 
    11271096\label{sec:DYN_ldf} 
     
    11611130} 
    11621131 
    1163 % ================================================================ 
    1164 \subsection[Iso-level laplacian (\forcode{ln_dynldf_lap})]{Iso-level laplacian operator (\protect\np{ln_dynldf_lap}{ln\_dynldf\_lap})} 
    1165 \label{subsec:DYN_ldf_lap} 
    1166  
    1167 For lateral iso-level diffusion, the discrete operator is: 
    1168 \begin{equation} 
    1169   \label{eq:DYN_ldf_lap} 
    1170   \left\{ 
    1171     \begin{aligned} 
    1172       D_u^{l{\mathrm {\mathbf U}}} =\frac{1}{e_{1u} }\delta_{i+1/2} \left[ {A_T^{lm} 
    1173           \;\chi } \right]-\frac{1}{e_{2u} {\kern 1pt}e_{3u} }\delta_j \left[ 
    1174         {A_f^{lm} \;e_{3f} \zeta } \right] \\ \\ 
    1175       D_v^{l{\mathrm {\mathbf U}}} =\frac{1}{e_{2v} }\delta_{j+1/2} \left[ {A_T^{lm} 
    1176           \;\chi } \right]+\frac{1}{e_{1v} {\kern 1pt}e_{3v} }\delta_i \left[ 
    1177         {A_f^{lm} \;e_{3f} \zeta } \right] 
    1178     \end{aligned} 
    1179   \right. 
    1180 \end{equation} 
    1181  
    1182 As explained in \autoref{subsec:MB_ldf}, 
    1183 this formulation (as the gradient of a divergence and curl of the vorticity) preserves symmetry and 
    1184 ensures a complete separation between the vorticity and divergence parts of the momentum diffusion. 
    1185  
    1186 %-------------------------------------------------------------------------------------------------------------- 
    1187 %           Rotated laplacian operator 
    1188 %-------------------------------------------------------------------------------------------------------------- 
    1189 \subsection[Rotated laplacian (\forcode{ln_dynldf_iso})]{Rotated laplacian operator (\protect\np{ln_dynldf_iso}{ln\_dynldf\_iso})} 
    1190 \label{subsec:DYN_ldf_iso} 
    1191  
    1192 A rotation of the lateral momentum diffusion operator is needed in several cases: 
    1193 for iso-neutral diffusion in the $z$-coordinate (\np[=.true.]{ln_dynldf_iso}{ln\_dynldf\_iso}) and 
    1194 for either iso-neutral (\np[=.true.]{ln_dynldf_iso}{ln\_dynldf\_iso}) or 
    1195 geopotential (\np[=.true.]{ln_dynldf_hor}{ln\_dynldf\_hor}) diffusion in the $s$-coordinate. 
    1196 In the partial step case, coordinates are horizontal except at the deepest level and 
    1197 no rotation is performed when \np[=.true.]{ln_dynldf_hor}{ln\_dynldf\_hor}. 
    1198 The diffusion operator is defined simply as the divergence of down gradient momentum fluxes on 
    1199 each momentum component. 
    1200 It must be emphasized that this formulation ignores constraints on the stress tensor such as symmetry. 
    1201 The resulting discrete representation is: 
    1202 \begin{equation} 
    1203   \label{eq:DYN_ldf_iso} 
    1204   \begin{split} 
    1205     D_u^{l\textbf{U}} &= \frac{1}{e_{1u} \, e_{2u} \, e_{3u} } \\ 
    1206     &  \left\{\quad  {\delta_{i+1/2} \left[ {A_T^{lm}  \left( 
    1207               {\frac{e_{2t} \; e_{3t} }{e_{1t} } \,\delta_{i}[u] 
    1208                 -e_{2t} \; r_{1t} \,\overline{\overline {\delta_{k+1/2}[u]}}^{\,i,\,k}} 
    1209             \right)} \right]}    \right. \\ 
    1210     & \qquad +\ \delta_j \left[ {A_f^{lm} \left( {\frac{e_{1f}\,e_{3f} }{e_{2f} 
    1211             }\,\delta_{j+1/2} [u] - e_{1f}\, r_{2f} 
    1212             \,\overline{\overline {\delta_{k+1/2} [u]}} ^{\,j+1/2,\,k}} 
    1213         \right)} \right] \\ 
    1214     &\qquad +\ \delta_k \left[ {A_{uw}^{lm} \left( {-e_{2u} \, r_{1uw} \,\overline{\overline 
    1215               {\delta_{i+1/2} [u]}}^{\,i+1/2,\,k+1/2} } 
    1216         \right.} \right. \\ 
    1217     &  \ \qquad \qquad \qquad \quad\ 
    1218     - e_{1u} \, r_{2uw} \,\overline{\overline {\delta_{j+1/2} [u]}} ^{\,j,\,k+1/2} \\ 
    1219     & \left. {\left. { \ \qquad \qquad \qquad \ \ \ \left. {\ 
    1220                 +\frac{e_{1u}\, e_{2u} }{e_{3uw} }\,\left( {r_{1uw}^2+r_{2uw}^2} 
    1221                 \right)\,\delta_{k+1/2} [u]} \right)} \right]\;\;\;} \right\} \\ \\ 
    1222     D_v^{l\textbf{V}} &= \frac{1}{e_{1v} \, e_{2v} \, e_{3v} } \\ 
    1223     &  \left\{\quad  {\delta_{i+1/2} \left[ {A_f^{lm}  \left( 
    1224               {\frac{e_{2f} \; e_{3f} }{e_{1f} } \,\delta_{i+1/2}[v] 
    1225                 -e_{2f} \; r_{1f} \,\overline{\overline {\delta_{k+1/2}[v]}}^{\,i+1/2,\,k}} 
    1226             \right)} \right]}    \right. \\ 
    1227     & \qquad +\ \delta_j \left[ {A_T^{lm} \left( {\frac{e_{1t}\,e_{3t} }{e_{2t} 
    1228             }\,\delta_{j} [v] - e_{1t}\, r_{2t} 
    1229             \,\overline{\overline {\delta_{k+1/2} [v]}} ^{\,j,\,k}} 
    1230         \right)} \right] \\ 
    1231     & \qquad +\ \delta_k \left[ {A_{vw}^{lm} \left( {-e_{2v} \, r_{1vw} \,\overline{\overline 
    1232               {\delta_{i+1/2} [v]}}^{\,i+1/2,\,k+1/2} }\right.} \right. \\ 
    1233     &  \ \qquad \qquad \qquad \quad\ 
    1234     - e_{1v} \, r_{2vw} \,\overline{\overline {\delta_{j+1/2} [v]}} ^{\,j+1/2,\,k+1/2} \\ 
    1235     & \left. {\left. { \ \qquad \qquad \qquad \ \ \ \left. {\ 
    1236                 +\frac{e_{1v}\, e_{2v} }{e_{3vw} }\,\left( {r_{1vw}^2+r_{2vw}^2} 
    1237                 \right)\,\delta_{k+1/2} [v]} \right)} \right]\;\;\;} \right\} 
    1238   \end{split} 
    1239 \end{equation} 
    1240 where $r_1$ and $r_2$ are the slopes between the surface along which the diffusion operator acts and 
    1241 the surface of computation ($z$- or $s$-surfaces). 
    1242 The way these slopes are evaluated is given in the lateral physics chapter (\autoref{chap:LDF}). 
    1243  
    1244 %-------------------------------------------------------------------------------------------------------------- 
    1245 %           Iso-level bilaplacian operator 
    1246 %-------------------------------------------------------------------------------------------------------------- 
    1247 \subsection[Iso-level bilaplacian (\forcode{ln_dynldf_bilap})]{Iso-level bilaplacian operator (\protect\np{ln_dynldf_bilap}{ln\_dynldf\_bilap})} 
    1248 \label{subsec:DYN_ldf_bilap} 
    1249  
    1250 The lateral fourth order operator formulation on momentum is obtained by applying \autoref{eq:DYN_ldf_lap} twice. 
    1251 It requires an additional assumption on boundary conditions: 
    1252 the first derivative term normal to the coast depends on the free or no-slip lateral boundary conditions chosen, 
    1253 while the third derivative terms normal to the coast are set to zero (see \autoref{chap:LBC}). 
    1254 %%% 
    1255 \gmcomment{add a remark on the the change in the position of the coefficient} 
    1256 %%% 
    1257  
    1258 % ================================================================ 
    12591132%           Vertical diffusion term 
    1260 % ================================================================ 
    1261 \section[Vertical diffusion term (\textit{dynzdf.F90})]{Vertical diffusion term (\protect\mdl{dynzdf})} 
    1262 \label{sec:DYN_zdf} 
    1263 %----------------------------------------------namzdf------------------------------------------------------ 
    1264  
    1265 %------------------------------------------------------------------------------------------------------------- 
    1266  
    1267 Options are defined through the \nam{zdf}{zdf} namelist variables. 
    1268 The large vertical diffusion coefficient found in the surface mixed layer together with high vertical resolution implies that in the case of explicit time stepping there would be too restrictive a constraint on the time step. 
    1269 Two time stepping schemes can be used for the vertical diffusion term: 
    1270 $(a)$ a forward time differencing scheme 
    1271 (\np[=.true.]{ln_zdfexp}{ln\_zdfexp}) using a time splitting technique (\np{nn_zdfexp}{nn\_zdfexp} $>$ 1) or 
    1272 $(b)$ a backward (or implicit) time differencing scheme (\np[=.false.]{ln_zdfexp}{ln\_zdfexp}) 
    1273 (see \autoref{chap:TD}). 
    1274 Note that namelist variables \np{ln_zdfexp}{ln\_zdfexp} and \np{nn_zdfexp}{nn\_zdfexp} apply to both tracers and dynamics. 
    1275  
    1276 The formulation of the vertical subgrid scale physics is the same whatever the vertical coordinate is. 
    1277 The vertical diffusion operators given by \autoref{eq:MB_zdf} take the following semi-discrete space form: 
    1278 \[ 
    1279   % \label{eq:DYN_zdf} 
    1280   \left\{ 
    1281     \begin{aligned} 
    1282       D_u^{vm} &\equiv \frac{1}{e_{3u}} \ \delta_k \left[ \frac{A_{uw}^{vm} }{e_{3uw} } 
    1283         \ \delta_{k+1/2} [\,u\,]         \right]     \\ 
    1284       \\ 
    1285       D_v^{vm} &\equiv \frac{1}{e_{3v}} \ \delta_k \left[ \frac{A_{vw}^{vm} }{e_{3vw} } 
    1286         \ \delta_{k+1/2} [\,v\,]         \right] 
    1287     \end{aligned} 
    1288   \right. 
    1289 \] 
    1290 where $A_{uw}^{vm} $ and $A_{vw}^{vm} $ are the vertical eddy viscosity and diffusivity coefficients. 
    1291 The way these coefficients are evaluated depends on the vertical physics used (see \autoref{chap:ZDF}). 
    1292  
    1293 The surface boundary condition on momentum is the stress exerted by the wind. 
    1294 At the surface, the momentum fluxes are prescribed as the boundary condition on 
    1295 the vertical turbulent momentum fluxes, 
    1296 \begin{equation} 
    1297   \label{eq:DYN_zdf_sbc} 
    1298   \left.{\left( {\frac{A^{vm} }{e_3 }\ \frac{\partial \textbf{U}_h}{\partial k}} \right)} \right|_{z=1} 
    1299   = \frac{1}{\rho_o} \binom{\tau_u}{\tau_v } 
    1300 \end{equation} 
    1301 where $\left( \tau_u ,\tau_v \right)$ are the two components of the wind stress vector in 
    1302 the (\textbf{i},\textbf{j}) coordinate system. 
    1303 The high mixing coefficients in the surface mixed layer ensure that the surface wind stress is distributed in 
    1304 the vertical over the mixed layer depth. 
    1305 If the vertical mixing coefficient is small (when no mixed layer scheme is used) 
    1306 the surface stress enters only the top model level, as a body force. 
    1307 The surface wind stress is calculated in the surface module routines (SBC, see \autoref{chap:SBC}). 
    1308  
    1309 The turbulent flux of momentum at the bottom of the ocean is specified through a bottom friction parameterisation 
    1310 (see \autoref{sec:ZDF_drg}) 
    1311  
    1312 % ================================================================ 
    13131133% External Forcing 
    1314 % ================================================================ 
    1315 \section{External forcings} 
    1316 \label{sec:DYN_forcing} 
    1317  
    1318 Besides the surface and bottom stresses (see the above section) 
    1319 which are introduced as boundary conditions on the vertical mixing, 
    1320 three other forcings may enter the dynamical equations by affecting the surface pressure gradient. 
    1321  
    1322 (1) When \np[=.true.]{ln_apr_dyn}{ln\_apr\_dyn} (see \autoref{sec:SBC_apr}), 
    1323 the atmospheric pressure is taken into account when computing the surface pressure gradient. 
    1324  
    1325 (2) When \np[=.true.]{ln_tide_pot}{ln\_tide\_pot} and \np[=.true.]{ln_tide}{ln\_tide} (see \autoref{sec:SBC_tide}), 
    1326 the tidal potential is taken into account when computing the surface pressure gradient. 
    1327  
    1328 (3) When \np[=2]{nn_ice_embd}{nn\_ice\_embd} and LIM or CICE is used 
    1329 (\ie\ when the sea-ice is embedded in the ocean), 
    1330 the snow-ice mass is taken into account when computing the surface pressure gradient. 
    1331  
    1332  
    1333 \gmcomment{ missing : the lateral boundary condition !!!   another external forcing 
    1334  } 
    1335  
    1336 % ================================================================ 
    13371134% Wetting and drying 
    1338 % ================================================================ 
    1339 \section{Wetting and drying } 
    1340 \label{sec:DYN_wetdry} 
    1341  
    1342 There are two main options for wetting and drying code (wd): 
    1343 (a) an iterative limiter (il) and (b) a directional limiter (dl). 
    1344 The directional limiter is based on the scheme developed by \cite{warner.defne.ea_CG13} for RO 
    1345 MS 
    1346 which was in turn based on ideas developed for POM by \cite{oey_OM06}. The iterative 
    1347 limiter is a new scheme.  The iterative limiter is activated by setting $\mathrm{ln\_wd\_il} = \mathrm{.true.}$ 
    1348 and $\mathrm{ln\_wd\_dl} = \mathrm{.false.}$. The directional limiter is activated 
    1349 by setting $\mathrm{ln\_wd\_dl} = \mathrm{.true.}$ and $\mathrm{ln\_wd\_il} = \mathrm{.false.}$. 
    1350  
    1351 \begin{listing} 
    1352   \nlst{namwad} 
    1353   \caption{\forcode{&namwad}} 
    1354   \label{lst:namwad} 
    1355 \end{listing} 
    1356  
    1357 The following terminology is used. The depth of the topography (positive downwards) 
    1358 at each $(i,j)$ point is the quantity stored in array $\mathrm{ht\_wd}$ in the \NEMO\ code. 
    1359 The height of the free surface (positive upwards) is denoted by $ \mathrm{ssh}$. Given the sign 
    1360 conventions used, the water depth, $h$, is the height of the free surface plus the depth of the 
    1361 topography (i.e. $\mathrm{ssh} + \mathrm{ht\_wd}$). 
    1362  
    1363 Both wd schemes take all points in the domain below a land elevation of $\mathrm{rn\_wdld}$ to be 
    1364 covered by water. They require the topography specified with a model 
    1365 configuration to have negative depths at points where the land is higher than the 
    1366 topography's reference sea-level. The vertical grid in \NEMO\ is normally computed relative to an 
    1367 initial state with zero sea surface height elevation. 
    1368 The user can choose to compute the vertical grid and heights in the model relative to 
    1369 a non-zero reference height for the free surface. This choice affects the calculation of the metrics and depths 
    1370 (i.e. the $\mathrm{e3t\_0, ht\_0}$ etc. arrays). 
    1371  
    1372 Points where the water depth is less than $\mathrm{rn\_wdmin1}$ are interpreted as ``dry''. 
    1373 $\mathrm{rn\_wdmin1}$ is usually chosen to be of order $0.05$m but extreme topographies 
    1374 with very steep slopes require larger values for normal choices of time-step. Surface fluxes 
    1375 are also switched off for dry cells to prevent freezing, boiling etc. of very thin water layers. 
    1376 The fluxes are tappered down using a $\mathrm{tanh}$ weighting function 
    1377 to no flux as the dry limit $\mathrm{rn\_wdmin1}$ is approached. Even wet cells can be very shallow. 
    1378 The depth at which to start tapering is controlled by the user by setting $\mathrm{rn\_wd\_sbcdep}$. 
    1379 The fraction $(<1)$ of sufrace fluxes to use at this depth is set by $\mathrm{rn\_wd\_sbcfra}$. 
    1380  
    1381 Both versions of the code have been tested in six test cases provided in the WAD\_TEST\_CASES configuration 
    1382 and in ``realistic'' configurations covering parts of the north-west European shelf. 
    1383 All these configurations have used pure sigma coordinates. It is expected that 
    1384 the wetting and drying code will work in domains with more general s-coordinates provided 
    1385 the coordinates are pure sigma in the region where wetting and drying actually occurs. 
    1386  
    1387 The next sub-section descrbies the directional limiter and the following sub-section the iterative limiter. 
    1388 The final sub-section covers some additional considerations that are relevant to both schemes. 
    1389  
    1390  
    1391 %----------------------------------------------------------------------------------------- 
    1392 %   Iterative limiters 
    1393 %----------------------------------------------------------------------------------------- 
    1394 \subsection[Directional limiter (\textit{wet\_dry.F90})]{Directional limiter (\mdl{wet\_dry})} 
    1395 \label{subsec:DYN_wd_directional_limiter} 
    1396  
    1397 The principal idea of the directional limiter is that 
    1398 water should not be allowed to flow out of a dry tracer cell (i.e. one whose water depth is less than \np{rn_wdmin1}{rn\_wdmin1}). 
    1399  
    1400 All the changes associated with this option are made to the barotropic solver for the non-linear 
    1401 free surface code within dynspg\_ts. 
    1402 On each barotropic sub-step the scheme determines the direction of the flow across each face of all the tracer cells 
    1403 and sets the flux across the face to zero when the flux is from a dry tracer cell. This prevents cells 
    1404 whose depth is rn\_wdmin1 or less from drying out further. The scheme does not force $h$ (the water depth) at tracer cells 
    1405 to be at least the minimum depth and hence is able to conserve mass / volume. 
    1406  
    1407 The flux across each $u$-face of a tracer cell is multiplied by a factor zuwdmask (an array which depends on ji and jj). 
    1408 If the user sets \np[=.false.]{ln_wd_dl_ramp}{ln\_wd\_dl\_ramp} then zuwdmask is 1 when the 
    1409 flux is from a cell with water depth greater than \np{rn_wdmin1}{rn\_wdmin1} and 0 otherwise. If the user sets 
    1410 \np[=.true.]{ln_wd_dl_ramp}{ln\_wd\_dl\_ramp} the flux across the face is ramped down as the water depth decreases 
    1411 from 2 * \np{rn_wdmin1}{rn\_wdmin1} to \np{rn_wdmin1}{rn\_wdmin1}. The use of this ramp reduced grid-scale noise in idealised test cases. 
    1412  
    1413 At the point where the flux across a $u$-face is multiplied by zuwdmask , we have chosen 
    1414 also to multiply the corresponding velocity on the ``now'' step at that face by zuwdmask. We could have 
    1415 chosen not to do that and to allow fairly large velocities to occur in these ``dry'' cells. 
    1416 The rationale for setting the velocity to zero is that it is the momentum equations that are being solved 
    1417 and the total momentum of the upstream cell (treating it as a finite volume) should be considered 
    1418 to be its depth times its velocity. This depth is considered to be zero at ``dry'' $u$-points consistent with its 
    1419 treatment in the calculation of the flux of mass across the cell face. 
    1420  
    1421  
    1422 \cite{warner.defne.ea_CG13} state that in their scheme the velocity masks at the cell faces for the baroclinic 
    1423 timesteps are set to 0 or 1 depending on whether the average of the masks over the barotropic sub-steps is respectively less than 
    1424 or greater than 0.5. That scheme does not conserve tracers in integrations started from constant tracer 
    1425 fields (tracers independent of $x$, $y$ and $z$). Our scheme conserves constant tracers because 
    1426 the velocities used at the tracer cell faces on the baroclinic timesteps are carefully calculated by dynspg\_ts 
    1427 to equal their mean value during the barotropic steps. If the user sets \np[=.true.]{ln_wd_dl_bc}{ln\_wd\_dl\_bc}, the 
    1428 baroclinic velocities are also multiplied by a suitably weighted average of zuwdmask. 
    1429  
    1430 %----------------------------------------------------------------------------------------- 
    1431 %   Iterative limiters 
    1432 %----------------------------------------------------------------------------------------- 
    1433  
    1434 \subsection[Iterative limiter (\textit{wet\_dry.F90})]{Iterative limiter (\mdl{wet\_dry})} 
    1435 \label{subsec:DYN_wd_iterative_limiter} 
    1436  
    1437 \subsubsection[Iterative flux limiter (\textit{wet\_dry.F90})]{Iterative flux limiter (\mdl{wet\_dry})} 
    1438 \label{subsec:DYN_wd_il_spg_limiter} 
    1439  
    1440 The iterative limiter modifies the fluxes across the faces of cells that are either already ``dry'' 
    1441 or may become dry within the next time-step using an iterative method. 
    1442  
    1443 The flux limiter for the barotropic flow (devised by Hedong Liu) can be understood as follows: 
    1444  
    1445 The continuity equation for the total water depth in a column 
    1446 \begin{equation} 
    1447   \label{eq:DYN_wd_continuity} 
    1448   \frac{\partial h}{\partial t} + \mathbf{\nabla.}(h\mathbf{u}) = 0 . 
    1449 \end{equation} 
    1450 can be written in discrete form  as 
    1451  
    1452 \begin{align} 
    1453   \label{eq:DYN_wd_continuity_2} 
    1454   \frac{e_1 e_2}{\Delta t} ( h_{i,j}(t_{n+1}) - h_{i,j}(t_e) ) 
    1455   &= - ( \mathrm{flxu}_{i+1,j} - \mathrm{flxu}_{i,j}  + \mathrm{flxv}_{i,j+1} - \mathrm{flxv}_{i,j} ) \\ 
    1456   &= \mathrm{zzflx}_{i,j} . 
    1457 \end{align} 
    1458  
    1459 In the above $h$ is the depth of the water in the column at point $(i,j)$, 
    1460 $\mathrm{flxu}_{i+1,j}$ is the flux out of the ``eastern'' face of the cell and 
    1461 $\mathrm{flxv}_{i,j+1}$ the flux out of the ``northern'' face of the cell; $t_{n+1}$ is 
    1462 the new timestep, $t_e$ is the old timestep (either $t_b$ or $t_n$) and $ \Delta t = 
    1463 t_{n+1} - t_e$; $e_1 e_2$ is the area of the tracer cells centred at $(i,j)$ and 
    1464 $\mathrm{zzflx}$ is the sum of the fluxes through all the faces. 
    1465  
    1466 The flux limiter splits the flux $\mathrm{zzflx}$ into fluxes that are out of the cell 
    1467 (zzflxp) and fluxes that are into the cell (zzflxn).  Clearly 
    1468  
    1469 \begin{equation} 
    1470   \label{eq:DYN_wd_zzflx_p_n_1} 
    1471   \mathrm{zzflx}_{i,j} = \mathrm{zzflxp}_{i,j} + \mathrm{zzflxn}_{i,j} . 
    1472 \end{equation} 
    1473  
    1474 The flux limiter iteratively adjusts the fluxes $\mathrm{flxu}$ and $\mathrm{flxv}$ until 
    1475 none of the cells will ``dry out''. To be precise the fluxes are limited until none of the 
    1476 cells has water depth less than $\mathrm{rn\_wdmin1}$ on step $n+1$. 
    1477  
    1478 Let the fluxes on the $m$th iteration step be denoted by $\mathrm{flxu}^{(m)}$ and 
    1479 $\mathrm{flxv}^{(m)}$.  Then the adjustment is achieved by seeking a set of coefficients, 
    1480 $\mathrm{zcoef}_{i,j}^{(m)}$ such that: 
    1481  
    1482 \begin{equation} 
    1483   \label{eq:DYN_wd_continuity_coef} 
    1484   \begin{split} 
    1485     \mathrm{zzflxp}^{(m)}_{i,j} =& \mathrm{zcoef}_{i,j}^{(m)} \mathrm{zzflxp}^{(0)}_{i,j} \\ 
    1486     \mathrm{zzflxn}^{(m)}_{i,j} =& \mathrm{zcoef}_{i,j}^{(m)} \mathrm{zzflxn}^{(0)}_{i,j} 
    1487   \end{split} 
    1488 \end{equation} 
    1489  
    1490 where the coefficients are $1.0$ generally but can vary between $0.0$ and $1.0$ around 
    1491 cells that would otherwise dry. 
    1492  
    1493 The iteration is initialised by setting 
    1494  
    1495 \begin{equation} 
    1496   \label{eq:DYN_wd_zzflx_initial} 
    1497   \mathrm{zzflxp^{(0)}}_{i,j} = \mathrm{zzflxp}_{i,j} , \quad  \mathrm{zzflxn^{(0)}}_{i,j} = \mathrm{zzflxn}_{i,j} . 
    1498 \end{equation} 
    1499  
    1500 The fluxes out of cell $(i,j)$ are updated at the $m+1$th iteration if the depth of the 
    1501 cell 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 (\autoref{eq:DYN_wd_continuity_2}) this 
    1503 condition is 
    1504  
    1505 \begin{equation} 
    1506   \label{eq:DYN_wd_continuity_if} 
    1507   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} ) . 
    1508 \end{equation} 
    1509  
    1510 Rearranging (\autoref{eq:DYN_wd_continuity_if}) we can obtain an expression for the maximum 
    1511 outward flux that can be allowed and still maintain the minimum wet depth: 
    1512  
    1513 \begin{equation} 
    1514   \label{eq:DYN_wd_max_flux} 
    1515   \begin{split} 
    1516     \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{]} \\ 
    1517     \phantom{[} & -  \mathrm{zzflxn}^{(m)}_{i,j} \Big] 
    1518   \end{split} 
    1519 \end{equation} 
    1520  
    1521 Note a small tolerance ($\mathrm{rn\_wdmin2}$) has been introduced here {\itshape [Q: Why is 
    1522 this necessary/desirable?]}. Substituting from (\autoref{eq:DYN_wd_continuity_coef}) gives an 
    1523 expression for the coefficient needed to multiply the outward flux at this cell in order 
    1524 to avoid drying. 
    1525  
    1526 \begin{equation} 
    1527   \label{eq:DYN_wd_continuity_nxtcoef} 
    1528   \begin{split} 
    1529     \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{]} \\ 
    1530     \phantom{[} & -  \mathrm{zzflxn}^{(m)}_{i,j} \Big] \frac{1}{ \mathrm{zzflxp}^{(0)}_{i,j} } 
    1531   \end{split} 
    1532 \end{equation} 
    1533  
    1534 Only the outward flux components are altered but, of course, outward fluxes from one cell 
    1535 are inward fluxes to adjacent cells and the balance in these cells may need subsequent 
    1536 adjustment; hence the iterative nature of this scheme.  Note, for example, that the flux 
    1537 across the ``eastern'' face of the $(i,j)$th cell is only updated at the $m+1$th iteration 
    1538 if that flux at the $m$th iteration is out of the $(i,j)$th cell. If that is the case then 
    1539 the flux across that face is into the $(i+1,j)$ cell and that flux will not be updated by 
    1540 the calculation for the $(i+1,j)$th cell. In this sense the updates to the fluxes across 
    1541 the faces of the cells do not ``compete'' (they do not over-write each other) and one 
    1542 would expect the scheme to converge relatively quickly. The scheme is flux based so 
    1543 conserves mass. It also conserves constant tracers for the same reason that the 
    1544 directional limiter does. 
    1545  
    1546  
    1547 %---------------------------------------------------------------------------------------- 
    1548 %      Surface pressure gradients 
    1549 %---------------------------------------------------------------------------------------- 
    1550 \subsubsection[Modification of surface pressure gradients (\textit{dynhpg.F90})]{Modification of surface pressure gradients (\mdl{dynhpg})} 
    1551 \label{subsec:DYN_wd_il_spg} 
    1552  
    1553 At ``dry'' points the water depth is usually close to $\mathrm{rn\_wdmin1}$. If the 
    1554 topography is sloping at these points the sea-surface will have a similar slope and there 
    1555 will hence be very large horizontal pressure gradients at these points. The WAD modifies 
    1556 the magnitude but not the sign of the surface pressure gradients (zhpi and zhpj) at such 
    1557 points by mulitplying them by positive factors (zcpx and zcpy respectively) that lie 
    1558 between $0$ and $1$. 
    1559  
    1560 We describe how the scheme works for the ``eastward'' pressure gradient, zhpi, calculated 
    1561 at the $(i,j)$th $u$-point. The scheme uses the ht\_wd depths and surface heights at the 
    1562 neighbouring $(i+1,j)$ and $(i,j)$ tracer points.  zcpx is calculated using two logicals 
    1563 variables, $\mathrm{ll\_tmp1}$ and $\mathrm{ll\_tmp2}$ which are evaluated for each grid 
    1564 column.  The three possible combinations are illustrated in \autoref{fig:DYN_WAD_dynhpg}. 
    1565  
    1566 %>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> 
    1567 \begin{figure}[!ht] 
    1568   \centering 
    1569   \includegraphics[width=0.66\textwidth]{Fig_WAD_dynhpg} 
    1570   \caption[Combinations controlling the limiting of the horizontal pressure gradient in 
    1571   wetting and drying regimes]{ 
    1572     Three possible combinations of the logical variables controlling the 
    1573     limiting of the horizontal pressure gradient in wetting and drying regimes} 
    1574   \label{fig:DYN_WAD_dynhpg} 
    1575 \end{figure} 
    1576 %>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> 
    1577  
    1578 The first logical, $\mathrm{ll\_tmp1}$, is set to true if and only if the water depth at 
    1579 both neighbouring points is greater than $\mathrm{rn\_wdmin1} + \mathrm{rn\_wdmin2}$ and 
    1580 the minimum height of the sea surface at the two points is greater than the maximum height 
    1581 of the topography at the two points: 
    1582  
    1583 \begin{equation} 
    1584   \label{eq:DYN_ll_tmp1} 
    1585   \begin{split} 
    1586     \mathrm{ll\_tmp1}  = & \mathrm{MIN(sshn(ji,jj), sshn(ji+1,jj))} > \\ 
    1587                      & \quad \mathrm{MAX(-ht\_wd(ji,jj), -ht\_wd(ji+1,jj))\  .and.} \\ 
    1588                      & \mathrm{MAX(sshn(ji,jj) + ht\_wd(ji,jj),} \\ 
    1589                      & \mathrm{\phantom{MAX(}sshn(ji+1,jj) + ht\_wd(ji+1,jj))} >\\ 
    1590                      & \quad\quad\mathrm{rn\_wdmin1 + rn\_wdmin2 } 
    1591   \end{split} 
    1592 \end{equation} 
    1593  
    1594 The second logical, $\mathrm{ll\_tmp2}$, is set to true if and only if the maximum height 
    1595 of the sea surface at the two points is greater than the maximum height of the topography 
    1596 at the two points plus $\mathrm{rn\_wdmin1} + \mathrm{rn\_wdmin2}$ 
    1597  
    1598 \begin{equation} 
    1599   \label{eq:DYN_ll_tmp2} 
    1600   \begin{split} 
    1601     \mathrm{ ll\_tmp2 } = & \mathrm{( ABS( sshn(ji,jj) - sshn(ji+1,jj) ) > 1.E-12 )\ .AND.}\\ 
    1602     & \mathrm{( MAX(sshn(ji,jj), sshn(ji+1,jj)) > } \\ 
    1603     & \mathrm{\phantom{(} MAX(-ht\_wd(ji,jj), -ht\_wd(ji+1,jj)) + rn\_wdmin1 + rn\_wdmin2}) . 
    1604   \end{split} 
    1605 \end{equation} 
    1606  
    1607 If $\mathrm{ll\_tmp1}$ is true then the surface pressure gradient, zhpi at the $(i,j)$ 
    1608 point is unmodified. If both logicals are false zhpi is set to zero. 
    1609  
    1610 If $\mathrm{ll\_tmp1}$ is true and $\mathrm{ll\_tmp2}$ is false then the surface pressure 
    1611 gradient is multiplied through by zcpx which is the absolute value of the difference in 
    1612 the water depths at the two points divided by the difference in the surface heights at the 
    1613 two points. Thus the sign of the sea surface height gradient is retained but the magnitude 
    1614 of the pressure force is determined by the difference in water depths rather than the 
    1615 difference in surface height between the two points. Note that dividing by the difference 
    1616 between the sea surface heights can be problematic if the heights approach parity. An 
    1617 additional condition is applied to $\mathrm{ ll\_tmp2 }$ to ensure it is .false. in such 
    1618 conditions. 
    1619  
    1620 \subsubsection[Additional considerations (\textit{usrdef\_zgr.F90})]{Additional considerations (\mdl{usrdef\_zgr})} 
    1621 \label{subsec:DYN_WAD_additional} 
    1622  
    1623 In the very shallow water where wetting and drying occurs the parametrisation of 
    1624 bottom drag is clearly very important. In order to promote stability 
    1625 it is sometimes useful to calculate the bottom drag using an implicit time-stepping approach. 
    1626  
    1627 Suitable specifcation of the surface heat flux in wetting and drying domains in forced and 
    1628 coupled simulations needs further consideration. In order to prevent freezing or boiling 
    1629 in uncoupled integrations the net surface heat fluxes need to be appropriately limited. 
    1630  
    1631 %---------------------------------------------------------------------------------------- 
    1632 %      The WAD test cases 
    1633 %---------------------------------------------------------------------------------------- 
    1634 \subsection[The WAD test cases (\textit{usrdef\_zgr.F90})]{The WAD test cases (\mdl{usrdef\_zgr})} 
    1635 \label{subsec:DYN_WAD_test_cases} 
    1636  
    1637 See the WAD tests MY\_DOC documention for details of the WAD test cases. 
    1638  
    1639  
    1640  
    1641 % ================================================================ 
    16421135% Time evolution term 
    1643 % ================================================================ 
    1644 \section[Time evolution term (\textit{dynnxt.F90})]{Time evolution term (\protect\mdl{dynnxt})} 
    1645 \label{sec:DYN_nxt} 
    1646  
    1647 %----------------------------------------------namdom---------------------------------------------------- 
    1648  
    1649 %------------------------------------------------------------------------------------------------------------- 
    1650  
    1651 Options are defined through the \nam{dom}{dom} namelist variables. 
    1652 The general framework for dynamics time stepping is a leap-frog scheme, 
    1653 \ie\ a three level centred time scheme associated with an Asselin time filter (cf. \autoref{chap:TD}). 
    1654 The scheme is applied to the velocity, except when 
    1655 using the flux form of momentum advection (cf. \autoref{sec:DYN_adv_cor_flux}) 
    1656 in the variable volume case (\texttt{vvl?} defined), 
    1657 where it has to be applied to the thickness weighted velocity (see \autoref{sec:SCOORD_momentum}) 
    1658  
    1659 $\bullet$ vector invariant form or linear free surface 
    1660 (\np[=.true.]{ln_dynhpg_vec}{ln\_dynhpg\_vec} ; \texttt{vvl?} not defined): 
    1661 \[ 
    1662   % \label{eq:DYN_nxt_vec} 
    1663   \left\{ 
    1664     \begin{aligned} 
    1665       &u^{t+\rdt} = u_f^{t-\rdt} + 2\rdt  \ \text{RHS}_u^t     \\ 
    1666       &u_f^t \;\quad = u^t+\gamma \,\left[ {u_f^{t-\rdt} -2u^t+u^{t+\rdt}} \right] 
    1667     \end{aligned} 
    1668   \right. 
    1669 \] 
    1670  
    1671 $\bullet$ flux form and nonlinear free surface 
    1672 (\np[=.false.]{ln_dynhpg_vec}{ln\_dynhpg\_vec} ; \texttt{vvl?} defined): 
    1673 \[ 
    1674   % \label{eq:DYN_nxt_flux} 
    1675   \left\{ 
    1676     \begin{aligned} 
    1677       &\left(e_{3u}\,u\right)^{t+\rdt} = \left(e_{3u}\,u\right)_f^{t-\rdt} + 2\rdt \; e_{3u} \;\text{RHS}_u^t     \\ 
    1678       &\left(e_{3u}\,u\right)_f^t \;\quad = \left(e_{3u}\,u\right)^t 
    1679       +\gamma \,\left[ {\left(e_{3u}\,u\right)_f^{t-\rdt} -2\left(e_{3u}\,u\right)^t+\left(e_{3u}\,u\right)^{t+\rdt}} \right] 
    1680     \end{aligned} 
    1681   \right. 
    1682 \] 
    1683 where RHS is the right hand side of the momentum equation, 
    1684 the subscript $f$ denotes filtered values and $\gamma$ is the Asselin coefficient. 
    1685 $\gamma$ is initialized as \np{nn_atfp}{nn\_atfp} (namelist parameter). 
    1686 Its default value is \np[=10.e-3]{nn_atfp}{nn\_atfp}. 
    1687 In both cases, the modified Asselin filter is not applied since perfect conservation is not an issue for 
    1688 the momentum equations. 
    1689  
    1690 Note that with the filtered free surface, 
    1691 the update of the \textit{after} velocities is done in the \mdl{dynsp\_flt} module, 
    1692 and only array swapping and Asselin filtering is done in \mdl{dynnxt}. 
    1693  
    1694 \onlyinsubfile{\input{../../global/epilogue}} 
    1695  
    1696 \end{document} 
  • NEMO/trunk/doc/latex/NEMO/subfiles/chap_LBC.tex

    r11584 r11596  
    22 
    33\begin{document} 
    4 % ================================================================ 
    5 % Chapter — Lateral Boundary Condition (LBC) 
    6 % ================================================================ 
    74\chapter{Lateral Boundary Condition (LBC)} 
    85\label{chap:LBC} 
     
    107\chaptertoc 
    118 
    12 \newpage 
    13  
    149%gm% add here introduction to this chapter 
    1510 
    16 % ================================================================ 
    17 % Boundary Condition at the Coast 
    18 % ================================================================ 
    1911\section[Boundary condition at the coast (\forcode{rn_shlat})]{Boundary condition at the coast (\protect\np{rn_shlat}{rn\_shlat})} 
    2012\label{sec:LBC_coast} 
     
    6860(normal velocity $u$ remains zero at the coast) (\autoref{fig:LBC_uv}). 
    6961 
    70 %>>>>>>>>>>>>>>>>>>>>>>>>>>>> 
    7162\begin{figure}[!t] 
    7263  \centering 
     
    7768  \label{fig:LBC_uv} 
    7869\end{figure} 
    79 %>>>>>>>>>>>>>>>>>>>>>>>>>>>> 
    8070 
    8171For momentum the situation is a bit more complex as two boundary conditions must be provided along the coast 
     
    9585These are: 
    9686 
    97 %>>>>>>>>>>>>>>>>>>>>>>>>>>>> 
    9887\begin{figure}[!p] 
    9988  \centering 
     
    10897  \label{fig:LBC_shlat} 
    10998\end{figure} 
    110 %>>>>>>>>>>>>>>>>>>>>>>>>>>>> 
    11199 
    112100\begin{description} 
    113101 
    114 \item[free-slip boundary condition ({\np[=0]{rn_shlat}{rn\_shlat}})] the tangential velocity at 
     102\item [free-slip boundary condition ({\np[=0]{rn_shlat}{rn\_shlat}})] the tangential velocity at 
    115103  the coastline is equal to the offshore velocity, 
    116104  \ie\ the normal derivative of the tangential velocity is zero at the coast, 
     
    118106  (\autoref{fig:LBC_shlat}-a). 
    119107 
    120 \item[no-slip boundary condition ({\np[=2]{rn_shlat}{rn\_shlat}})] the tangential velocity vanishes at the coastline. 
     108\item [no-slip boundary condition ({\np[=2]{rn_shlat}{rn\_shlat}})] the tangential velocity vanishes at the coastline. 
    121109  Assuming that the tangential velocity decreases linearly from 
    122110  the closest ocean velocity grid point to the coastline, 
     
    139127  \] 
    140128 
    141 \item["partial" free-slip boundary condition (0$<$\np{rn_shlat}{rn\_shlat}$<$2)] the tangential velocity at 
     129\item ["partial" free-slip boundary condition (0$<$\np{rn_shlat}{rn\_shlat}$<$2)] the tangential velocity at 
    142130  the coastline is smaller than the offshore velocity, \ie\ there is a lateral friction but 
    143131  not strong enough to make the tangential velocity at the coast vanish (\autoref{fig:LBC_shlat}-c). 
    144132  This can be selected by providing a value of mask$_{f}$ strictly inbetween $0$ and $2$. 
    145133 
    146 \item["strong" no-slip boundary condition (2$<$\np{rn_shlat}{rn\_shlat})] the viscous boundary layer is assumed to 
     134\item ["strong" no-slip boundary condition (2$<$\np{rn_shlat}{rn\_shlat})] the viscous boundary layer is assumed to 
    147135  be smaller than half the grid size (\autoref{fig:LBC_shlat}-d). 
    148136  The friction is thus larger than in the no-slip case. 
     
    154142it is only applied next to the coast where the minimum water depth can be quite shallow. 
    155143 
    156  
    157 % ================================================================ 
    158 % Boundary Condition around the Model Domain 
    159 % ================================================================ 
    160144\section[Model domain boundary condition (\forcode{jperio})]{Model domain boundary condition (\protect\jp{jperio})} 
    161145\label{sec:LBC_jperio} 
     
    166150The north-fold boundary condition is associated with the 3-pole ORCA mesh. 
    167151 
    168 % ------------------------------------------------------------------------------------------------------------- 
    169 %        Closed, cyclic (\jp{jperio}\forcode{ = 0..2}) 
    170 % ------------------------------------------------------------------------------------------------------------- 
    171152\subsection[Closed, cyclic (\forcode{=0,1,2,7})]{Closed, cyclic (\protect\jp{jperio}\forcode{=0,1,2,7})} 
    172153\label{subsec:LBC_jperio012} 
     
    182163\begin{description} 
    183164 
    184 \item[For closed boundary (\jp{jperio}\forcode{=0})], 
     165\item [For closed boundary (\jp{jperio}\forcode{=0})], 
    185166  solid walls are imposed at all model boundaries: 
    186167  first and last rows and columns are set to zero. 
    187168 
    188 \item[For cyclic east-west boundary (\jp{jperio}\forcode{=1})], 
     169\item [For cyclic east-west boundary (\jp{jperio}\forcode{=1})], 
    189170  first and last rows are set to zero (closed) whilst the first column is set to 
    190171  the value of the last-but-one column and the last column to the value of the second one 
     
    192173  Whatever flows out of the eastern (western) end of the basin enters the western (eastern) end. 
    193174 
    194 \item[For cyclic north-south boundary (\jp{jperio}\forcode{=2})], 
     175\item [For cyclic north-south boundary (\jp{jperio}\forcode{=2})], 
    195176  first and last columns are set to zero (closed) whilst the first row is set to 
    196177  the value of the last-but-one row and the last row to the value of the second one 
     
    198179  Whatever flows out of the northern (southern) end of the basin enters the southern (northern) end. 
    199180 
    200 \item[Bi-cyclic east-west and north-south boundary (\jp{jperio}\forcode{=7})] combines cases 1 and 2. 
     181\item [Bi-cyclic east-west and north-south boundary (\jp{jperio}\forcode{=7})] combines cases 1 and 2. 
    201182 
    202183\end{description} 
    203184 
    204 %>>>>>>>>>>>>>>>>>>>>>>>>>>>> 
    205185\begin{figure}[!t] 
    206186  \centering 
     
    210190  \label{fig:LBC_jperio} 
    211191\end{figure} 
    212 %>>>>>>>>>>>>>>>>>>>>>>>>>>>> 
    213  
    214 % ------------------------------------------------------------------------------------------------------------- 
    215 %        North fold (\textit{jperio = 3 }to $6)$ 
    216 % ------------------------------------------------------------------------------------------------------------- 
     192 
    217193\subsection[North-fold (\forcode{=3,6})]{North-fold (\protect\jp{jperio}\forcode{=3,6})} 
    218194\label{subsec:LBC_north_fold} 
     
    224200Further information can be found in \mdl{lbcnfd} module which applies the north fold boundary condition. 
    225201 
    226 %>>>>>>>>>>>>>>>>>>>>>>>>>>>> 
    227202\begin{figure}[!t] 
    228203  \centering 
     
    234209  \label{fig:LBC_North_Fold_T} 
    235210\end{figure} 
    236 %>>>>>>>>>>>>>>>>>>>>>>>>>>>> 
    237  
    238 % ==================================================================== 
    239 % Exchange with neighbouring processors 
    240 % ==================================================================== 
     211 
    241212\section[Exchange with neighbouring processors (\textit{lbclnk.F90}, \textit{lib\_mpp.F90})]{Exchange with neighbouring processors (\protect\mdl{lbclnk}, \protect\mdl{lib\_mpp})} 
    242213\label{sec:LBC_mpp} 
     
    284255many communications during 1 time step of the model.\\ 
    285256 
    286 %>>>>>>>>>>>>>>>>>>>>>>>>>>>> 
    287257\begin{figure}[!t] 
    288258  \centering 
     
    291261  \label{fig:LBC_mpp} 
    292262\end{figure} 
    293 %>>>>>>>>>>>>>>>>>>>>>>>>>>>> 
    294263 
    295264In \NEMO, the splitting is regular and arithmetic. 
     
    339308When land processors are eliminated, the value corresponding to these locations in the model output files is undefined. \np{ln_mskland}{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}{nn\_msh}). 
    340309 
    341 %>>>>>>>>>>>>>>>>>>>>>>>>>>>> 
    342310\begin{figure}[!ht] 
    343311  \centering 
     
    352320  \label{fig:LBC_mppini2} 
    353321\end{figure} 
    354 %>>>>>>>>>>>>>>>>>>>>>>>>>>>> 
    355  
    356  
    357 % ==================================================================== 
    358 % Unstructured open boundaries BDY 
    359 % ==================================================================== 
     322 
    360323\section{Unstructured open boundary conditions (BDY)} 
    361324\label{sec:LBC_bdy} 
     
    411374 
    412375\begin{description} 
    413 \item[\forcode{'none'}:] No boundary condition applied. 
     376\item [\forcode{'none'}:] No boundary condition applied. 
    414377  So the solution will ``see'' the land points around the edge of the edge of the domain. 
    415 \item[\forcode{'specified'}:] Specified boundary condition applied (only available for baroclinic velocity and tracer variables). 
    416 \item[\forcode{'neumann'}:] Value at the boundary are duplicated (No gradient). Only available for baroclinic velocity and tracer variables. 
    417 \item[\forcode{'frs'}:] Flow Relaxation Scheme (FRS) available for all variables. 
    418 \item[\forcode{'Orlanski'}:] Orlanski radiation scheme (fully oblique) for barotropic, baroclinic and tracer variables. 
    419 \item[\forcode{'Orlanski_npo'}:] Orlanski radiation scheme for barotropic, baroclinic and tracer variables. 
    420 \item[\forcode{'flather'}:] Flather radiation scheme for the barotropic variables only. 
     378\item [\forcode{'specified'}:] Specified boundary condition applied (only available for baroclinic velocity and tracer variables). 
     379\item [\forcode{'neumann'}:] Value at the boundary are duplicated (No gradient). Only available for baroclinic velocity and tracer variables. 
     380\item [\forcode{'frs'}:] Flow Relaxation Scheme (FRS) available for all variables. 
     381\item [\forcode{'Orlanski'}:] Orlanski radiation scheme (fully oblique) for barotropic, baroclinic and tracer variables. 
     382\item [\forcode{'Orlanski_npo'}:] Orlanski radiation scheme for barotropic, baroclinic and tracer variables. 
     383\item [\forcode{'flather'}:] Flather radiation scheme for the barotropic variables only. 
    421384\end{description} 
    422385 
     
    620583Only one mask file is used even if multiple boundary sets are defined. 
    621584 
    622 %>>>>>>>>>>>>>>>>>>>>>>>>>>>> 
    623585\begin{figure}[!t] 
    624586  \centering 
     
    627589  \label{fig:LBC_bdy_geom} 
    628590\end{figure} 
    629 %>>>>>>>>>>>>>>>>>>>>>>>>>>>> 
    630591 
    631592%---------------------------------------------- 
     
    657618will re-order the data in old BDY data files. 
    658619 
    659 %>>>>>>>>>>>>>>>>>>>>>>>>>>>> 
    660620\begin{figure}[!t] 
    661621  \centering 
     
    665625  \label{fig:LBC_nc_header} 
    666626\end{figure} 
    667 %>>>>>>>>>>>>>>>>>>>>>>>>>>>> 
    668627 
    669628%---------------------------------------------- 
  • NEMO/trunk/doc/latex/NEMO/subfiles/chap_LDF.tex

    r11584 r11596  
    33\begin{document} 
    44 
    5 % ================================================================ 
    6 % Chapter Lateral Ocean Physics (LDF) 
    7 % ================================================================ 
    85\chapter{Lateral Ocean Physics (LDF)} 
    96\label{chap:LDF} 
    107 
    118\chaptertoc 
    12  
    13 \newpage 
    149 
    1510The lateral physics terms in the momentum and tracer equations have been described in \autoref{eq:MB_zdf} and 
     
    3126%-------------------------------------------------------------------------------------------------------------- 
    3227 
    33 % ================================================================ 
    34 % Lateral Mixing Operator 
    35 % ================================================================ 
    3628\section[Lateral mixing operators]{Lateral mixing operators} 
    3729\label{sec:LDF_op} 
     
    5547We stress again that from \NEMO\ 4, the simultaneous use Laplacian and Bilaplacian operators is not allowed. 
    5648 
    57 % ================================================================ 
    58 % Direction of lateral Mixing 
    59 % ================================================================ 
    6049\section[Direction of lateral mixing (\textit{ldfslp.F90})]{Direction of lateral mixing (\protect\mdl{ldfslp})} 
    6150\label{sec:LDF_slp} 
     
    151140\begin{description} 
    152141 
    153 \item[$z$-coordinate with full step: ] 
     142\item [$z$-coordinate with full step: ] 
    154143  in \autoref{eq:LDF_slp_iso} the densities appearing in the $i$ and $j$ derivatives  are taken at the same depth, 
    155144  thus the $in situ$ density can be used. 
     
    158147  (see \autoref{subsec:TRA_bn2}). 
    159148 
    160 \item[$z$-coordinate with partial step: ] 
     149\item [$z$-coordinate with partial step: ] 
    161150  this case is identical to the full step case except that at partial step level, 
    162151  the \emph{horizontal} density gradient is evaluated as described in \autoref{sec:TRA_zpshde}. 
    163152 
    164 \item[$s$- or hybrid $s$-$z$- coordinate: ] 
     153\item [$s$- or hybrid $s$-$z$- coordinate: ] 
    165154  in the current release of \NEMO, iso-neutral mixing is only employed for $s$-coordinates if 
    166155  the Griffies scheme is used (\np[=.true.]{ln_traldf_triad}{ln\_traldf\_triad}; 
     
    234223contrary to the \citet{griffies.gnanadesikan.ea_JPO98} operator which has that property. 
    235224 
    236 %>>>>>>>>>>>>>>>>>>>>>>>>>>>> 
    237225\begin{figure}[!ht] 
    238226  \centering 
     
    241229  \label{fig:LDF_ZDF1} 
    242230\end{figure} 
    243 %>>>>>>>>>>>>>>>>>>>>>>>>>>>> 
    244231 
    245232%There are three additional questions about the slope calculation. 
     
    249236 
    250237%from griffies: chapter 13.1.... 
    251  
    252  
    253238 
    254239% In addition and also for numerical stability reasons \citep{cox_OM87, griffies_bk04}, 
     
    264249\colorbox{yellow}{The way slopes are tapered has be checked. Not sure that this is still what is actually done.} 
    265250 
    266 %>>>>>>>>>>>>>>>>>>>>>>>>>>>> 
    267251\begin{figure}[!ht] 
    268252  \centering 
     
    286270  \label{fig:LDF_eiv_slp} 
    287271\end{figure} 
    288 %>>>>>>>>>>>>>>>>>>>>>>>>>>>> 
    289272 
    290273\colorbox{yellow}{add here a discussion about the flattening of the slopes, vs tapering the coefficient.} 
     
    316299(see \autoref{sec:LBC_coast}). 
    317300 
    318  
    319 % ================================================================ 
    320 % Lateral Mixing Coefficients 
    321 % ================================================================ 
    322301\section[Lateral mixing coefficient (\forcode{nn_aht_ijk_t} \& \forcode{nn_ahm_ijk_t})]{Lateral mixing coefficient (\protect\np{nn_aht_ijk_t}{nn\_aht\_ijk\_t} \& \protect\np{nn_ahm_ijk_t}{nn\_ahm\_ijk\_t})} 
    323302\label{sec:LDF_coef} 
     
    467446(\autoref{sec:INVARIANTS_dynldf_properties}). 
    468447 
    469 % ================================================================ 
    470 % Eddy Induced Mixing 
    471 % ================================================================ 
    472448\section[Eddy induced velocity (\forcode{ln_ldfeiv})]{Eddy induced velocity (\protect\np{ln_ldfeiv}{ln\_ldfeiv})} 
    473449 
     
    483459 
    484460%-------------------------------------------------------------------------------------------------------------- 
    485  
    486461 
    487462%%gm  from Triad appendix  : to be incorporated.... 
     
    538513In case of setting \np[=.true.]{ln_traldf_triad}{ln\_traldf\_triad}, a skew form of the eddy induced advective fluxes is used, which is described in \autoref{apdx:TRIADS}. 
    539514 
    540 % ================================================================ 
    541 % Mixed layer eddies 
    542 % ================================================================ 
    543515\section[Mixed layer eddies (\forcode{ln_mle})]{Mixed layer eddies (\protect\np{ln_mle}{ln\_mle})} 
    544516\label{sec:LDF_mle} 
  • NEMO/trunk/doc/latex/NEMO/subfiles/chap_OBS.tex

    r11584 r11596  
    22 
    33\begin{document} 
    4 % ================================================================ 
    5 % Chapter observation operator (OBS) 
    6 % ================================================================ 
    74\chapter{Observation and Model Comparison (OBS)} 
    85\label{chap:OBS} 
     
    2219\end{figure} 
    2320 
    24 \newpage 
    25  
    2621The observation and model comparison code, the observation operator (OBS), reads in observation files 
    2722(profile temperature and salinity, sea surface temperature, sea level anomaly, sea ice concentration, and velocity) and calculates an interpolated model equivalent value at the observation location and nearest model time step. 
     
    6055In \autoref{sec:OBS_obsutils} we describe some utilities to help work with the files produced by the OBS code. 
    6156 
    62 % ================================================================ 
    63 % Example 
    64 % ================================================================ 
    6557\section{Running the observation operator code example} 
    6658\label{sec:OBS_example} 
     
    610602\begin{enumerate} 
    611603 
    612 \item[1.] {\bfseries Great-Circle distance-weighted interpolation.} 
     604\item [1.] {\bfseries Great-Circle distance-weighted interpolation.} 
    613605  The weights are computed as a function of the great-circle distance $s(P, \cdot)$ between $P$ and 
    614606  the model grid points $A$, $B$ etc. 
     
    655647   \end{alignat*} 
    656648 
    657 \item[2.] {\bfseries Great-Circle distance-weighted interpolation with small angle approximation.} 
     649\item [2.] {\bfseries Great-Circle distance-weighted interpolation with small angle approximation.} 
    658650  Similar to the previous interpolation but with the distance $s$ computed as 
    659651  \begin{alignat*}{2} 
     
    665657  where $M$ corresponds to $A$, $B$, $C$ or $D$. 
    666658 
    667 \item[3.] {\bfseries Bilinear interpolation for a regular spaced grid.} 
     659\item [3.] {\bfseries Bilinear interpolation for a regular spaced grid.} 
    668660  The interpolation is split into two 1D interpolations in the longitude and latitude directions, respectively. 
    669661 
    670 \item[4.] {\bfseries Bilinear remapping interpolation for a general grid.} 
     662\item [4.] {\bfseries Bilinear remapping interpolation for a general grid.} 
    671663  An iterative scheme that involves first mapping a quadrilateral cell into 
    672664  a cell with coordinates (0,0), (1,0), (0,1) and (1,1). 
     
    697689\autoref{fig:OBS_avgrec} and~\autoref{fig:OBS_avgrad}. 
    698690 
    699 %>>>>>>>>>>>>>>>>>>>>>>>>>>>> 
    700691\begin{figure} 
    701692  \centering 
     
    708699% >>>>>>>>>>>>>>>>>>>>>>>>>>>> 
    709700 
    710 %>>>>>>>>>>>>>>>>>>>>>>>>>>>> 
    711701\begin{figure} 
    712702  \centering 
     
    717707  \label{fig:OBS_avgrad} 
    718708\end{figure} 
    719 %>>>>>>>>>>>>>>>>>>>>>>>>>>>> 
    720  
    721709 
    722710\subsection{Grid search} 
     
    786774\subsubsection{Geographical distribution of observations among processors} 
    787775 
    788 %>>>>>>>>>>>>>>>>>>>>>>>>>>>> 
    789776\begin{figure} 
    790777  \centering 
     
    795782  \label{fig:OBS_local} 
    796783\end{figure} 
    797 %>>>>>>>>>>>>>>>>>>>>>>>>>>>> 
    798784 
    799785This is the simplest option in which the observations are distributed according to 
     
    814800\subsubsection{Round-robin distribution of observations among processors} 
    815801 
    816 %>>>>>>>>>>>>>>>>>>>>>>>>>>>> 
    817802\begin{figure} 
    818803  \centering 
     
    823808  \label{fig:OBS_global} 
    824809\end{figure} 
    825 %>>>>>>>>>>>>>>>>>>>>>>>>>>>> 
    826810 
    827811An alternative approach is to distribute the observations equally among processors and 
     
    843827 
    844828For profile observation types we do both vertical and horizontal interpolation. \NEMO\ has a generalised vertical coordinate system this means the vertical level depths can vary with location. Therefore, it is necessary first to perform vertical interpolation of the model value to the observation depths for each of the four surrounding grid points. After this the model values, at these points, at the observation depth, are horizontally interpolated to the observation location. 
    845  
    846 \newpage 
    847  
    848 % ================================================================ 
    849 % Standalone observation operator documentation 
    850 % ================================================================ 
    851829 
    852830%\usepackage{framed} 
     
    957935climatologies with the same set of observations. 
    958936This approach is referred to as \emph{Class 4} since it is the fourth metric defined by the GODAE intercomparison project. This requires multiple runs of the SAO and running an additional utility (not currently in the \NEMO\ repository) to combine the feedback files into one class 4 file. 
    959  
    960 \newpage 
    961937 
    962938\section{Observation utilities} 
     
    11461122The rightmost group of buttons will print the plot window as a postscript, save it as png, or exit from dataplot. 
    11471123 
    1148 %>>>>>>>>>>>>>>>>>>>>>>>>>>>> 
    11491124\begin{figure} 
    11501125  \centering 
     
    11531128  \label{fig:OBS_dataplotmain} 
    11541129\end{figure} 
    1155 %>>>>>>>>>>>>>>>>>>>>>>>>>>>> 
    11561130 
    11571131If a profile point is clicked with the mouse button a plot of the observation and background values as 
    11581132a function of depth (\autoref{fig:OBS_dataplotprofile}). 
    11591133 
    1160 %>>>>>>>>>>>>>>>>>>>>>>>>>>>> 
    11611134\begin{figure} 
    11621135  \centering 
     
    11661139  \label{fig:OBS_dataplotprofile} 
    11671140\end{figure} 
    1168 %>>>>>>>>>>>>>>>>>>>>>>>>>>>> 
    11691141 
    11701142\onlyinsubfile{\input{../../global/epilogue}} 
  • NEMO/trunk/doc/latex/NEMO/subfiles/chap_SBC.tex

    r11584 r11596  
    33\begin{document} 
    44 
    5 % ================================================================ 
    6 % Chapter —— Surface Boundary Condition (SBC, SAS, ISF, ICB) 
    7 % ================================================================ 
    85\chapter{Surface Boundary Condition (SBC, SAS, ISF, ICB)} 
    96\label{chap:SBC} 
    107 
    118\chaptertoc 
    12  
    13 \newpage 
    149 
    1510%---------------------------------------namsbc-------------------------------------------------- 
     
    2520 
    2621\begin{itemize} 
    27 \item 
    28   the two components of the surface ocean stress $\left( {\tau_u \;,\;\tau_v} \right)$ 
    29 \item 
    30   the incoming solar and non solar heat fluxes $\left( {Q_{ns} \;,\;Q_{sr} } \right)$ 
    31 \item 
    32   the surface freshwater budget $\left( {\textit{emp}} \right)$ 
    33 \item 
    34   the surface salt flux associated with freezing/melting of seawater $\left( {\textit{sfx}} \right)$ 
    35 \item 
    36   the atmospheric pressure at the ocean surface $\left( p_a \right)$ 
     22\item the two components of the surface ocean stress $\left( {\tau_u \;,\;\tau_v} \right)$ 
     23\item the incoming solar and non solar heat fluxes $\left( {Q_{ns} \;,\;Q_{sr} } \right)$ 
     24\item the surface freshwater budget $\left( {\textit{emp}} \right)$ 
     25\item the surface salt flux associated with freezing/melting of seawater $\left( {\textit{sfx}} \right)$ 
     26\item the atmospheric pressure at the ocean surface $\left( p_a \right)$ 
    3727\end{itemize} 
    3828 
     
    4131 
    4232\begin{itemize} 
    43 \item 
    44   a bulk formulation (\np[=.true.]{ln_blk}{ln\_blk} with four possible bulk algorithms), 
    45 \item 
    46   a flux formulation (\np[=.true.]{ln_flx}{ln\_flx}), 
    47 \item 
    48   a coupled or mixed forced/coupled formulation (exchanges with a atmospheric model via the OASIS coupler), 
     33\item a bulk formulation (\np[=.true.]{ln_blk}{ln\_blk} with four possible bulk algorithms), 
     34\item a flux formulation (\np[=.true.]{ln_flx}{ln\_flx}), 
     35\item a coupled or mixed forced/coupled formulation (exchanges with a atmospheric model via the OASIS coupler), 
    4936(\np{ln_cpl}{ln\_cpl} or \np[=.true.]{ln_mixcpl}{ln\_mixcpl}), 
    50 \item 
    51   a user defined formulation (\np[=.true.]{ln_usr}{ln\_usr}). 
     37\item a user defined formulation (\np[=.true.]{ln_usr}{ln\_usr}). 
    5238\end{itemize} 
    5339 
     
    6652 
    6753\begin{itemize} 
    68 \item 
    69   the rotation of vector components supplied relative to an east-north coordinate system onto 
     54\item the rotation of vector components supplied relative to an east-north coordinate system onto 
    7055  the local grid directions in the model, 
    71 \item 
    72   the use of a land/sea mask for input fields (\np[=.true.]{nn_lsm}{nn\_lsm}), 
    73 \item 
    74   the addition of a surface restoring term to observed SST and/or SSS (\np[=.true.]{ln_ssr}{ln\_ssr}), 
    75 \item 
    76   the modification of fluxes below ice-covered areas (using climatological ice-cover or a sea-ice model) 
     56\item the use of a land/sea mask for input fields (\np[=.true.]{nn_lsm}{nn\_lsm}), 
     57\item the addition of a surface restoring term to observed SST and/or SSS (\np[=.true.]{ln_ssr}{ln\_ssr}), 
     58\item the modification of fluxes below ice-covered areas (using climatological ice-cover or a sea-ice model) 
    7759  (\np[=0..3]{nn_ice}{nn\_ice}), 
    78 \item 
    79   the addition of river runoffs as surface freshwater fluxes or lateral inflow (\np[=.true.]{ln_rnf}{ln\_rnf}), 
    80 \item 
    81   the addition of ice-shelf melting as lateral inflow (parameterisation) or 
     60\item the addition of river runoffs as surface freshwater fluxes or lateral inflow (\np[=.true.]{ln_rnf}{ln\_rnf}), 
     61\item the addition of ice-shelf melting as lateral inflow (parameterisation) or 
    8262  as fluxes applied at the land-ice ocean interface (\np[=.true.]{ln_isf}{ln\_isf}), 
    83 \item 
    84   the addition of a freshwater flux adjustment in order to avoid a mean sea-level drift 
     63\item the addition of a freshwater flux adjustment in order to avoid a mean sea-level drift 
    8564  (\np[=0..2]{nn_fwb}{nn\_fwb}), 
    86 \item 
    87   the transformation of the solar radiation (if provided as daily mean) into an analytical diurnal cycle 
     65\item the transformation of the solar radiation (if provided as daily mean) into an analytical diurnal cycle 
    8866  (\np[=.true.]{ln_dm2dc}{ln\_dm2dc}), 
    89 \item 
    90   the activation of wave effects from an external wave model  (\np[=.true.]{ln_wave}{ln\_wave}), 
    91 \item 
    92   a neutral drag coefficient is read from an external wave model (\np[=.true.]{ln_cdgw}{ln\_cdgw}), 
    93 \item 
    94   the Stokes drift from an external wave model is accounted for (\np[=.true.]{ln_sdw}{ln\_sdw}), 
    95 \item 
    96   the choice of the Stokes drift profile parameterization (\np[=0..2]{nn_sdrift}{nn\_sdrift}), 
    97 \item 
    98   the surface stress given to the ocean is modified by surface waves (\np[=.true.]{ln_tauwoc}{ln\_tauwoc}), 
    99 \item 
    100   the surface stress given to the ocean is read from an external wave model (\np[=.true.]{ln_tauw}{ln\_tauw}), 
    101 \item 
    102   the Stokes-Coriolis term is included (\np[=.true.]{ln_stcor}{ln\_stcor}), 
    103 \item 
    104   the light penetration in the ocean (\np[=.true.]{ln_traqsr}{ln\_traqsr} with namelist \nam{tra_qsr}{tra\_qsr}), 
    105 \item 
    106   the atmospheric surface pressure gradient effect on ocean and ice dynamics (\np[=.true.]{ln_apr_dyn}{ln\_apr\_dyn} with namelist \nam{sbc_apr}{sbc\_apr}), 
    107 \item 
    108   the effect of sea-ice pressure on the ocean (\np[=.true.]{ln_ice_embd}{ln\_ice\_embd}). 
     67\item the activation of wave effects from an external wave model  (\np[=.true.]{ln_wave}{ln\_wave}), 
     68\item a neutral drag coefficient is read from an external wave model (\np[=.true.]{ln_cdgw}{ln\_cdgw}), 
     69\item the Stokes drift from an external wave model is accounted for (\np[=.true.]{ln_sdw}{ln\_sdw}), 
     70\item the choice of the Stokes drift profile parameterization (\np[=0..2]{nn_sdrift}{nn\_sdrift}), 
     71\item the surface stress given to the ocean is modified by surface waves (\np[=.true.]{ln_tauwoc}{ln\_tauwoc}), 
     72\item the surface stress given to the ocean is read from an external wave model (\np[=.true.]{ln_tauw}{ln\_tauw}), 
     73\item the Stokes-Coriolis term is included (\np[=.true.]{ln_stcor}{ln\_stcor}), 
     74\item the light penetration in the ocean (\np[=.true.]{ln_traqsr}{ln\_traqsr} with namelist \nam{tra_qsr}{tra\_qsr}), 
     75\item the atmospheric surface pressure gradient effect on ocean and ice dynamics (\np[=.true.]{ln_apr_dyn}{ln\_apr\_dyn} with namelist \nam{sbc_apr}{sbc\_apr}), 
     76\item the effect of sea-ice pressure on the ocean (\np[=.true.]{ln_ice_embd}{ln\_ice\_embd}). 
    10977\end{itemize} 
    11078 
     
    11987which provides additional sources of fresh water. 
    12088 
    121  
    122  
    123 % ================================================================ 
    124 % Surface boundary condition for the ocean 
    125 % ================================================================ 
    12689\section{Surface boundary condition for the ocean} 
    12790\label{sec:SBC_ocean} 
     
    155118$(ii)$ it changes the surface temperature and salinity through the heat and salt contents of 
    156119the mass exchanged with atmosphere, sea-ice and ice shelves. 
    157  
    158120 
    159121%\colorbox{yellow}{Miss: } 
     
    185147these averaged fields are used to compute the surface fluxes at the frequency of \np{nn_fsbc}{nn\_fsbc} time-steps. 
    186148 
    187  
    188149%-------------------------------------------------TABLE--------------------------------------------------- 
    189150\begin{table}[tb] 
     
    210171%\colorbox{yellow}{Penser a} mettre dans le restant l'info nn\_fsbc ET nn\_fsbc*rdt de sorte de reinitialiser la moyenne si on change la frequence ou le pdt 
    211172 
    212  
    213  
    214 % ================================================================ 
    215 %       Input Data 
    216 % ================================================================ 
    217173\section{Input data generic interface} 
    218174\label{sec:SBC_input} 
     
    223179The module is designed with four main objectives in mind: 
    224180\begin{enumerate} 
    225 \item 
    226   optionally provide a time interpolation of the input data every specified model time-step, whatever their input frequency is, 
     181\item optionally provide a time interpolation of the input data every specified model time-step, whatever their input frequency is, 
    227182  and according to the different calendars available in the model. 
    228 \item 
    229   optionally provide an on-the-fly space interpolation from the native input data grid to the model grid. 
    230 \item 
    231   make the run duration independent from the period cover by the input files. 
    232 \item 
    233   provide a simple user interface and a rather simple developer interface by 
     183\item optionally provide an on-the-fly space interpolation from the native input data grid to the model grid. 
     184\item make the run duration independent from the period cover by the input files. 
     185\item provide a simple user interface and a rather simple developer interface by 
    234186  limiting the number of prerequisite informations. 
    235187\end{enumerate} 
     
    251203By default, the data are assumed to be in the same directory as the executable, so that cn\_dir='./'. 
    252204 
    253  
    254 % ------------------------------------------------------------------------------------------------------------- 
    255 % Input Data specification (\mdl{fldread}) 
    256 % ------------------------------------------------------------------------------------------------------------- 
    257205\subsection[Input data specification (\textit{fldread.F90})]{Input data specification (\protect\mdl{fldread})} 
    258206\label{subsec:SBC_fldread} 
     
    265213where 
    266214\begin{description} 
    267 \item[File name]: 
     215\item [File name]: 
    268216  the stem name of the NetCDF file to be opened. 
    269217  This stem will be completed automatically by the model, with the addition of a '.nc' at its end and 
     
    301249%-------------------------------------------------------------------------------------------------------------- 
    302250 
    303  
    304 \item[Record frequency]: 
     251\item [Record frequency]: 
    305252  the frequency of the records contained in the input file. 
    306253  Its unit is in hours if it is positive (for example 24 for daily forcing) or in months if negative 
     
    309256  On some computers, setting it to '24.' can be interpreted as 240! 
    310257 
    311 \item[Variable name]: 
     258\item [Variable name]: 
    312259  the name of the variable to be read in the input NetCDF file. 
    313260 
    314 \item[Time interpolation]: 
     261\item [Time interpolation]: 
    315262  a logical to activate, or not, the time interpolation. 
    316263  If set to 'false', the forcing will have a steplike shape remaining constant during each forcing period. 
     
    322269  linear interpolation will be performed between mid-day of two consecutive days. 
    323270 
    324 \item[Climatological forcing]: 
     271\item [Climatological forcing]: 
    325272  a logical to specify if a input file contains climatological forcing which can be cycle in time, 
    326273  or an interannual forcing which will requires additional files if 
     
    328275  See the above file naming strategy which impacts the expected name of the file to be opened. 
    329276 
    330 \item[Open/close frequency]: 
     277\item [Open/close frequency]: 
    331278  the frequency at which forcing files must be opened/closed. 
    332279  Four cases are coded: 
     
    337284  the experiment is not starting at the beginning of the year. 
    338285 
    339 \item[Others]: 
     286\item [Others]: 
    340287  'weights filename', 'pairing rotation' and 'land/sea mask' are associated with 
    341288  on-the-fly interpolation which is described in \autoref{subsec:SBC_iof}. 
     
    378325a useful feature for user considering that it is too heavy to manipulate the complete file for year Y-1. 
    379326 
    380  
    381 % ------------------------------------------------------------------------------------------------------------- 
    382 % Interpolation on the Fly 
    383 % ------------------------------------------------------------------------------------------------------------- 
    384327\subsection{Interpolation on-the-fly} 
    385328\label{subsec:SBC_iof} 
     
    404347Note that nn\_lsm=0 forces the code to not apply the procedure, even if a land/sea mask file is supplied. 
    405348 
    406  
    407 % ------------------------------------------------------------------------------------------------------------- 
    408 % Bilinear interpolation 
    409 % ------------------------------------------------------------------------------------------------------------- 
    410349\subsubsection{Bilinear interpolation} 
    411350\label{subsec:SBC_iof_bilinear} 
     
    429368and wgt(1) corresponds to variable "wgt01" for example. 
    430369 
    431  
    432 % ------------------------------------------------------------------------------------------------------------- 
    433 % Bicubic interpolation 
    434 % ------------------------------------------------------------------------------------------------------------- 
    435370\subsubsection{Bicubic interpolation} 
    436371\label{subsec:SBC_iof_bicubic} 
     
    451386the spatial dependency has been included into the weights. 
    452387 
    453  
    454 % ------------------------------------------------------------------------------------------------------------- 
    455 % Implementation 
    456 % ------------------------------------------------------------------------------------------------------------- 
    457388\subsubsection{Implementation} 
    458389\label{subsec:SBC_iof_imp} 
     
    490421or is a copy of one from the first few columns on the opposite side of the grid (cyclical case). 
    491422 
    492  
    493 % ------------------------------------------------------------------------------------------------------------- 
    494 % Limitations 
    495 % ------------------------------------------------------------------------------------------------------------- 
    496423\subsubsection{Limitations} 
    497424\label{subsec:SBC_iof_lim} 
    498425 
    499426\begin{enumerate} 
    500 \item 
    501   The case where input data grids are not logically rectangular (irregular grid case) has not been tested. 
    502 \item 
    503   This code is not guaranteed to produce positive definite answers from positive definite inputs when 
     427\item The case where input data grids are not logically rectangular (irregular grid case) has not been tested. 
     428\item This code is not guaranteed to produce positive definite answers from positive definite inputs when 
    504429  a bicubic interpolation method is used. 
    505 \item 
    506   The cyclic condition is only applied on left and right columns, and not to top and bottom rows. 
    507 \item 
    508   The gradients across the ends of a cyclical grid assume that the grid spacing between 
     430\item The cyclic condition is only applied on left and right columns, and not to top and bottom rows. 
     431\item The gradients across the ends of a cyclical grid assume that the grid spacing between 
    509432  the two columns involved are consistent with the weights used. 
    510 \item 
    511   Neither interpolation scheme is conservative. (There is a conservative scheme available in SCRIP, 
     433\item Neither interpolation scheme is conservative. (There is a conservative scheme available in SCRIP, 
    512434  but this has not been implemented.) 
    513435\end{enumerate} 
     
    520442(see the directory NEMOGCM/TOOLS/WEIGHTS). 
    521443 
    522  
    523 % ------------------------------------------------------------------------------------------------------------- 
    524 % Standalone Surface Boundary Condition Scheme 
    525 % ------------------------------------------------------------------------------------------------------------- 
    526444\subsection{Standalone surface boundary condition scheme (SAS)} 
    527445\label{subsec:SBC_SAS} 
     
    541459 
    542460\begin{itemize} 
    543 \item 
    544   Multiple runs of the model are required in code development to 
     461\item Multiple runs of the model are required in code development to 
    545462  see the effect of different algorithms in the bulk formulae. 
    546 \item 
    547   The effect of different parameter sets in the ice model is to be examined. 
    548 \item 
    549   Development of sea-ice algorithms or parameterizations. 
    550 \item 
    551   Spinup of the iceberg floats 
    552 \item 
    553   Ocean/sea-ice simulation with both models running in parallel (\np[=.true.]{ln_mixcpl}{ln\_mixcpl}) 
     463\item The effect of different parameter sets in the ice model is to be examined. 
     464\item Development of sea-ice algorithms or parameterizations. 
     465\item Spinup of the iceberg floats 
     466\item Ocean/sea-ice simulation with both models running in parallel (\np[=.true.]{ln_mixcpl}{ln\_mixcpl}) 
    554467\end{itemize} 
    555468 
     
    562475 
    563476\begin{itemize} 
    564 \item 
    565   \mdl{nemogcm}: 
     477\item \mdl{nemogcm}: 
    566478  This routine initialises the rest of the model and repeatedly calls the stp time stepping routine (\mdl{step}). 
    567479  Since the ocean state is not calculated all associated initialisations have been removed. 
    568 \item 
    569   \mdl{step}: 
     480\item \mdl{step}: 
    570481  The main time stepping routine now only needs to call the sbc routine (and a few utility functions). 
    571 \item 
    572   \mdl{sbcmod}: 
     482\item \mdl{sbcmod}: 
    573483  This has been cut down and now only calculates surface forcing and the ice model required. 
    574484  New surface modules that can function when only the surface level of the ocean state is defined can also be added 
    575485  (\eg\ icebergs). 
    576 \item 
    577   \mdl{daymod}: 
     486\item \mdl{daymod}: 
    578487  No ocean restarts are read or written (though the ice model restarts are retained), 
    579488  so calls to restart functions have been removed. 
    580489  This also means that the calendar cannot be controlled by time in a restart file, 
    581490  so the user must check that nn\_date0 in the model namelist is correct for his or her purposes. 
    582 \item 
    583   \mdl{stpctl}: 
     491\item \mdl{stpctl}: 
    584492  Since there is no free surface solver, references to it have been removed from \rou{stp\_ctl} module. 
    585 \item 
    586   \mdl{diawri}: 
     493\item \mdl{diawri}: 
    587494  All 3D data have been removed from the output. 
    588495  The surface temperature, salinity and velocity components (which have been read in) are written along with 
     
    593500 
    594501\begin{itemize} 
    595 \item 
    596   \mdl{sbcsas}: 
     502\item \mdl{sbcsas}: 
    597503  This module initialises the input files needed for reading temperature, salinity and 
    598504  velocity arrays at the surface. 
     
    604510\end{itemize} 
    605511 
    606  
    607512The user can also choose in the \nam{sbc_sas}{sbc\_sas} namelist to read the mean (nn\_fsbc time-step) fraction of solar net radiation absorbed in the 1st T level using 
    608513 (\np[=.true.]{ln_flx}{ln\_flx}) and to provide 3D oceanic velocities instead of 2D ones (\np{ln_flx}{ln\_flx}\forcode{=.true.}). In that last case, only the 1st level will be read in. 
    609514 
    610  
    611  
    612 % ================================================================ 
    613 % Flux formulation 
    614 % ================================================================ 
    615515\section[Flux formulation (\textit{sbcflx.F90})]{Flux formulation (\protect\mdl{sbcflx})} 
    616516\label{sec:SBC_flx} 
     
    634534See \autoref{subsec:SBC_ssr} for its specification. 
    635535 
    636  
    637  
    638 % ================================================================ 
    639 % Bulk formulation 
    640 % ================================================================ 
    641536\section[Bulk formulation (\textit{sbcblk.F90})]{Bulk formulation (\protect\mdl{sbcblk})} 
    642537\label{sec:SBC_blk} 
     
    720615the namsbc\_blk namelist (see \autoref{subsec:SBC_fldread}). 
    721616 
    722  
    723 % ------------------------------------------------------------------------------------------------------------- 
    724 %        Ocean-Atmosphere Bulk formulae 
    725 % ------------------------------------------------------------------------------------------------------------- 
    726617\subsection[Ocean-Atmosphere Bulk formulae (\textit{sbcblk\_algo\_coare.F90, sbcblk\_algo\_coare3p5.F90, sbcblk\_algo\_ecmwf.F90, sbcblk\_algo\_ncar.F90})]{Ocean-Atmosphere Bulk formulae (\mdl{sbcblk\_algo\_coare}, \mdl{sbcblk\_algo\_coare3p5}, \mdl{sbcblk\_algo\_ecmwf}, \mdl{sbcblk\_algo\_ncar})} 
    727618\label{subsec:SBC_blk_ocean} 
     
    731622their neutral transfer coefficients relationships with neutral wind. 
    732623\begin{itemize} 
    733 \item 
    734   NCAR (\np[=.true.]{ln_NCAR}{ln\_NCAR}): 
     624\item NCAR (\np[=.true.]{ln_NCAR}{ln\_NCAR}): 
    735625  The NCAR bulk formulae have been developed by \citet{large.yeager_rpt04}. 
    736626  They have been designed to handle the NCAR forcing, a mixture of NCEP reanalysis and satellite data. 
     
    741631  Note that substituting ERA40 to NCEP reanalysis fields does not require changes in the bulk formulea themself. 
    742632  This is the so-called DRAKKAR Forcing Set (DFS) \citep{brodeau.barnier.ea_OM10}. 
    743 \item 
    744   COARE 3.0 (\np[=.true.]{ln_COARE_3p0}{ln\_COARE\_3p0}): 
     633\item COARE 3.0 (\np[=.true.]{ln_COARE_3p0}{ln\_COARE\_3p0}): 
    745634  See \citet{fairall.bradley.ea_JC03} for more details 
    746 \item 
    747   COARE 3.5 (\np[=.true.]{ln_COARE_3p5}{ln\_COARE\_3p5}): 
     635\item COARE 3.5 (\np[=.true.]{ln_COARE_3p5}{ln\_COARE\_3p5}): 
    748636  See \citet{edson.jampana.ea_JPO13} for more details 
    749 \item 
    750   ECMWF (\np[=.true.]{ln_ECMWF}{ln\_ECMWF}): 
     637\item ECMWF (\np[=.true.]{ln_ECMWF}{ln\_ECMWF}): 
    751638  Based on \href{https://www.ecmwf.int/node/9221}{IFS (Cy31)} implementation and documentation. 
    752639  Surface roughness lengths needed for the Obukhov length are computed following \citet{beljaars_QJRMS95}. 
    753640\end{itemize} 
    754641 
    755 % ------------------------------------------------------------------------------------------------------------- 
    756 %        Ice-Atmosphere Bulk formulae 
    757 % ------------------------------------------------------------------------------------------------------------- 
    758642\subsection{Ice-Atmosphere Bulk formulae} 
    759643\label{subsec:SBC_blk_ice} 
     
    762646 
    763647\begin{itemize} 
    764 \item 
    765   Constant value (\np[ Cd_ice=1.4e-3 ]{constant value}{constant\ value}): 
     648\item Constant value (\np[ Cd_ice=1.4e-3 ]{constant value}{constant\ value}): 
    766649  default constant value used for momentum and heat neutral transfer coefficients 
    767 \item 
    768   \citet{lupkes.gryanik.ea_JGR12} (\np[=.true.]{ln_Cd_L12}{ln\_Cd\_L12}): 
     650\item \citet{lupkes.gryanik.ea_JGR12} (\np[=.true.]{ln_Cd_L12}{ln\_Cd\_L12}): 
    769651  This scheme adds a dependency on edges at leads, melt ponds and flows 
    770652  of the constant neutral air-ice drag. After some approximations, 
     
    773655  starting at 1.5e-3 for A=0, reaching 1.97e-3 for A=0.5 and going down 1.4e-3 for A=1. 
    774656  It is theoretically applicable to all ice conditions (not only MIZ). 
    775 \item 
    776   \citet{lupkes.gryanik_JGR15} (\np[=.true.]{ln_Cd_L15}{ln\_Cd\_L15}): 
     657\item \citet{lupkes.gryanik_JGR15} (\np[=.true.]{ln_Cd_L15}{ln\_Cd\_L15}): 
    777658  Alternative turbulent transfer coefficients formulation between sea-ice 
    778659  and atmosphere with distinct momentum and heat coefficients depending 
     
    784665\end{itemize} 
    785666 
    786  
    787  
    788 % ================================================================ 
    789 % Coupled formulation 
    790 % ================================================================ 
    791667\section[Coupled formulation (\textit{sbccpl.F90})]{Coupled formulation (\protect\mdl{sbccpl})} 
    792668\label{sec:SBC_cpl} 
     
    826702In cases where this is definitely not possible, the model should abort with an error message. 
    827703 
    828  
    829  
    830 % ================================================================ 
    831 %        Atmospheric pressure 
    832 % ================================================================ 
    833704\section[Atmospheric pressure (\textit{sbcapr.F90})]{Atmospheric pressure (\protect\mdl{sbcapr})} 
    834705\label{sec:SBC_apr} 
     
    868739\np{ln_apr_obc}{ln\_apr\_obc}  might be set to true. 
    869740 
    870  
    871  
    872 % ================================================================ 
    873 %        Surface Tides Forcing 
    874 % ================================================================ 
    875741\section[Surface tides (\textit{sbctide.F90})]{Surface tides (\protect\mdl{sbctide})} 
    876742\label{sec:SBC_tide} 
     
    925791\forcode{.false.} removes the SAL contribution. 
    926792 
    927  
    928  
    929 % ================================================================ 
    930 %        River runoffs 
    931 % ================================================================ 
    932793\section[River runoffs (\textit{sbcrnf.F90})]{River runoffs (\protect\mdl{sbcrnf})} 
    933794\label{sec:SBC_rnf} 
     
    950811%required to properly represent the diurnal cycle \citep{bernie.woolnough.ea_JC05}. see also \autoref{fig:SBC_dcy}.}. 
    951812 
    952  
    953813%To do this we need to treat evaporation/precipitation fluxes and river runoff differently in the 
    954814%\mdl{tra\_sbc} module.  We decided to separate them throughout the code, so that the variable 
     
    957817%emp).  This meant many uses of emp and emps needed to be changed, a list of all modules which use 
    958818%emp or emps and the changes made are below: 
    959  
    960819 
    961820%Rachel: 
     
    1035894as the ocean water leaving the domain removes heat and salt (at the same concentration) with it. 
    1036895 
    1037  
    1038896%\colorbox{yellow}{Nevertheless, Pb of vertical resolution and 3D input : increase vertical mixing near river mouths to mimic a 3D river 
    1039897 
     
    1055913%To do this we need to treat evaporation/precipitation fluxes and river runoff differently in the tra_sbc module.  We decided to separate them throughout the code, so that the variable emp represented solely evaporation minus precipitation fluxes, and a new 2d variable rnf was added which represents the volume flux of river runoff (in kg/m2s to remain consistent with emp).  This meant many uses of emp and emps needed to be changed, a list of all modules which use emp or emps and the changes made are below: 
    1056914 
    1057  
    1058  
    1059 % ================================================================ 
    1060 %        Ice shelf melting 
    1061 % ================================================================ 
    1062915\section[Ice shelf melting (\textit{sbcisf.F90})]{Ice shelf melting (\protect\mdl{sbcisf})} 
    1063916\label{sec:SBC_isf} 
     
    1077930\begin{description} 
    1078931 
    1079   \item[{\np[=1]{nn_isf}{nn\_isf}}]: 
     932  \item [{\np[=1]{nn_isf}{nn\_isf}}]: 
    1080933  The ice shelf cavity is represented (\np[=.true.]{ln_isfcav}{ln\_isfcav} needed). 
    1081934  The fwf and heat flux are depending of the local water properties. 
     
    1084937 
    1085938   \begin{description} 
    1086    \item[{\np[=1]{nn_isfblk}{nn\_isfblk}}]: 
     939   \item [{\np[=1]{nn_isfblk}{nn\_isfblk}}]: 
    1087940     The melt rate is based on a balance between the upward ocean heat flux and 
    1088941     the latent heat flux at the ice shelf base. A complete description is available in \citet{hunter_rpt06}. 
    1089    \item[{\np[=2]{nn_isfblk}{nn\_isfblk}}]: 
     942   \item [{\np[=2]{nn_isfblk}{nn\_isfblk}}]: 
    1090943     The melt rate and the heat flux are based on a 3 equations formulation 
    1091944     (a heat flux budget at the ice base, a salt flux budget at the ice base and a linearised freezing point temperature equation). 
     
    1104957     There are 3 different ways to compute the exchange coeficient: 
    1105958   \begin{description} 
    1106         \item[{\np[=0]{nn_gammablk}{nn\_gammablk}}]: 
     959        \item [{\np[=0]{nn_gammablk}{nn\_gammablk}}]: 
    1107960     The salt and heat exchange coefficients are constant and defined by \np{rn_gammas0}{rn\_gammas0} and \np{rn_gammat0}{rn\_gammat0}. 
    1108961     \begin{gather*} 
     
    1112965     \end{gather*} 
    1113966     This is the recommended formulation for ISOMIP. 
    1114    \item[{\np[=1]{nn_gammablk}{nn\_gammablk}}]: 
     967   \item [{\np[=1]{nn_gammablk}{nn\_gammablk}}]: 
    1115968     The salt and heat exchange coefficients are velocity dependent and defined as 
    1116969     \begin{gather*} 
     
    1120973     where $u_{*}$ is the friction velocity in the top boundary layer (ie first \np{rn_hisf_tbl}{rn\_hisf\_tbl} meters). 
    1121974     See \citet{jenkins.nicholls.ea_JPO10} for all the details on this formulation. It is the recommended formulation for realistic application. 
    1122    \item[{\np[=2]{nn_gammablk}{nn\_gammablk}}]: 
     975   \item [{\np[=2]{nn_gammablk}{nn\_gammablk}}]: 
    1123976     The salt and heat exchange coefficients are velocity and stability dependent and defined as: 
    1124977\[ 
     
    1131984     This formulation has not been extensively tested in \NEMO\ (not recommended). 
    1132985   \end{description} 
    1133   \item[{\np[=2]{nn_isf}{nn\_isf}}]: 
     986  \item [{\np[=2]{nn_isf}{nn\_isf}}]: 
    1134987   The ice shelf cavity is not represented. 
    1135988   The fwf and heat flux are computed using the \citet{beckmann.goosse_OM03} parameterisation of isf melting. 
     
    1138991   (\np{sn_depmin_isf}{sn\_depmin\_isf}) as in (\np[=3]{nn_isf}{nn\_isf}). 
    1139992   The effective melting length (\np{sn_Leff_isf}{sn\_Leff\_isf}) is read from a file. 
    1140   \item[{\np[=3]{nn_isf}{nn\_isf}}]: 
     993  \item [{\np[=3]{nn_isf}{nn\_isf}}]: 
    1141994   The ice shelf cavity is not represented. 
    1142995   The fwf (\np{sn_rnfisf}{sn\_rnfisf}) is prescribed and distributed along the ice shelf edge between 
     
    1144997   the base of the ice shelf along the calving front (\np{sn_depmin_isf}{sn\_depmin\_isf}). 
    1145998   The heat flux ($Q_h$) is computed as $Q_h = fwf \times L_f$. 
    1146   \item[{\np[=4]{nn_isf}{nn\_isf}}]: 
     999  \item [{\np[=4]{nn_isf}{nn\_isf}}]: 
    11471000   The ice shelf cavity is opened (\np[=.true.]{ln_isfcav}{ln\_isfcav} needed). 
    11481001   However, the fwf is not computed but specified from file \np{sn_fwfisf}{sn\_fwfisf}). 
     
    11671020See the runoff section \autoref{sec:SBC_rnf} for all the details about the divergence correction.\\ 
    11681021 
    1169 %>>>>>>>>>>>>>>>>>>>>>>>>>>>> 
    11701022\begin{figure}[!t] 
    11711023  \centering 
     
    11761028  \label{fig:SBC_isf} 
    11771029\end{figure} 
    1178 %>>>>>>>>>>>>>>>>>>>>>>>>>>>> 
    1179  
    1180  
    1181  
    1182 % ================================================================ 
    1183 %        Ice sheet coupling 
    1184 % ================================================================ 
     1030 
    11851031\section{Ice sheet coupling} 
    11861032\label{sec:SBC_iscpl} 
     
    11981044 
    11991045\begin{description} 
    1200 \item[Step 1]: the ice sheet model send a new bathymetry and ice shelf draft netcdf file. 
    1201 \item[Step 2]: a new domcfg.nc file is built using the DOMAINcfg tools. 
    1202 \item[Step 3]: \NEMO\ run for a specific period and output the average melt rate over the period. 
    1203 \item[Step 4]: the ice sheet model run using the melt rate outputed in step 4. 
    1204 \item[Step 5]: go back to 1. 
     1046\item [Step 1]: the ice sheet model send a new bathymetry and ice shelf draft netcdf file. 
     1047\item [Step 2]: a new domcfg.nc file is built using the DOMAINcfg tools. 
     1048\item [Step 3]: \NEMO\ run for a specific period and output the average melt rate over the period. 
     1049\item [Step 4]: the ice sheet model run using the melt rate outputed in step 4. 
     1050\item [Step 5]: go back to 1. 
    12051051\end{description} 
    12061052 
     
    12101056 
    12111057\begin{description} 
    1212 \item[Thin a cell down]: 
     1058\item [Thin a cell down]: 
    12131059  T/S/ssh are unchanged and U/V in the top cell are corrected to keep the barotropic transport (bt) constant 
    12141060  ($bt_b=bt_n$). 
    1215 \item[Enlarge  a cell]: 
     1061\item [Enlarge  a cell]: 
    12161062  See case "Thin a cell down" 
    1217 \item[Dry a cell]: 
     1063\item [Dry a cell]: 
    12181064  mask, T/S, U/V and ssh are set to 0. 
    12191065  Furthermore, U/V into the water column are modified to satisfy ($bt_b=bt_n$). 
    1220 \item[Wet a cell]: 
     1066\item [Wet a cell]: 
    12211067  mask is set to 1, T/S is extrapolated from neighbours, $ssh_n = ssh_b$ and U/V set to 0. 
    12221068  If no neighbours, T/S is extrapolated from old top cell value. 
    12231069  If no neighbours along i,j and k (both previous test failed), T/S/U/V/ssh and mask are set to 0. 
    1224 \item[Dry a column]: 
     1070\item [Dry a column]: 
    12251071   mask, T/S, U/V are set to 0 everywhere in the column and ssh set to 0. 
    1226 \item[Wet a column]: 
     1072\item [Wet a column]: 
    12271073  set mask to 1, T/S is extrapolated from neighbours, ssh is extrapolated from neighbours and U/V set to 0. 
    12281074  If no neighbour, T/S/U/V and mask set to 0. 
     
    12471093The corrective increment is apply into the cell itself (if it is a wet cell), the neigbouring cells or the closest wet cell (if the cell is now dry). 
    12481094 
    1249  
    1250  
    1251 % ================================================================ 
    1252 %        Handling of icebergs 
    1253 % ================================================================ 
    12541095\section{Handling of icebergs (ICB)} 
    12551096\label{sec:SBC_ICB_icebergs} 
     
    12751116Two initialisation schemes are possible. 
    12761117\begin{description} 
    1277 \item[{\np{nn_test_icebergs}{nn\_test\_icebergs}~$>$~0}] 
     1118\item [{\np{nn_test_icebergs}{nn\_test\_icebergs}~$>$~0}] 
    12781119  In this scheme, the value of \np{nn_test_icebergs}{nn\_test\_icebergs} represents the class of iceberg to generate 
    12791120  (so between 1 and 10), and \np{nn_test_icebergs}{nn\_test\_icebergs} provides a lon/lat box in the domain at each grid point of 
     
    12821123  \np{nn_test_icebergs}{nn\_test\_icebergs} is defined by four numbers in \np{nn_test_box}{nn\_test\_box} representing the corners of 
    12831124  the geographical box: lonmin,lonmax,latmin,latmax 
    1284 \item[{\np[=-1]{nn_test_icebergs}{nn\_test\_icebergs}}] 
     1125\item [{\np[=-1]{nn_test_icebergs}{nn\_test\_icebergs}}] 
    12851126  In this scheme, the model reads a calving file supplied in the \np{sn_icb}{sn\_icb} parameter. 
    12861127  This should be a file with a field on the configuration grid (typically ORCA) 
     
    13071148The amount of information is controlled by two integer parameters: 
    13081149\begin{description} 
    1309 \item[{\np{nn_verbose_level}{nn\_verbose\_level}}] takes a value between one and four and 
     1150\item [{\np{nn_verbose_level}{nn\_verbose\_level}}] takes a value between one and four and 
    13101151  represents an increasing number of points in the code at which variables are written, 
    13111152  and an increasing level of obscurity. 
    1312 \item[{\np{nn_verbose_write}{nn\_verbose\_write}}] is the number of timesteps between writes 
     1153\item [{\np{nn_verbose_write}{nn\_verbose\_write}}] is the number of timesteps between writes 
    13131154\end{description} 
    13141155 
     
    13211162since its trajectory data may be spread across multiple files. 
    13221163 
    1323  
    1324  
    1325 % ============================================================================================================= 
    1326 %        Interactions with waves (sbcwave.F90, ln_wave) 
    1327 % ============================================================================================================= 
    13281164\section[Interactions with waves (\textit{sbcwave.F90}, \forcode{ln_wave})]{Interactions with waves (\protect\mdl{sbcwave}, \protect\np{ln_wave}{ln\_wave})} 
    13291165\label{sec:SBC_wave} 
     
    13491185Wave fields can be provided either in forced or coupled mode: 
    13501186\begin{description} 
    1351 \item[forced mode]: wave fields should be defined through the \nam{sbc_wave}{sbc\_wave} namelist 
     1187\item [forced mode]: wave fields should be defined through the \nam{sbc_wave}{sbc\_wave} namelist 
    13521188for external data names, locations, frequency, interpolation and all the miscellanous options allowed by 
    13531189Input Data generic Interface (see \autoref{sec:SBC_input}). 
    1354 \item[coupled mode]: \NEMO\ and an external wave model can be coupled by setting \np[=.true.]{ln_cpl}{ln\_cpl} 
     1190\item [coupled mode]: \NEMO\ and an external wave model can be coupled by setting \np[=.true.]{ln_cpl}{ln\_cpl} 
    13551191in \nam{sbc}{sbc} namelist and filling the \nam{sbc_cpl}{sbc\_cpl} namelist. 
    13561192\end{description} 
    1357  
    13581193 
    13591194% ---------------------------------------------------------------- 
     
    13691204the drag coefficient is computed according to the stable/unstable conditions of the 
    13701205air-sea interface following \citet{large.yeager_rpt04}. 
    1371  
    13721206 
    13731207% ---------------------------------------------------------------- 
     
    14091243 
    14101244\begin{description} 
    1411 \item[{\np{nn_sdrift}{nn\_sdrift} = 0}]: exponential integral profile parameterization proposed by 
     1245\item [{\np{nn_sdrift}{nn\_sdrift} = 0}]: exponential integral profile parameterization proposed by 
    14121246\citet{breivik.janssen.ea_JPO14}: 
    14131247 
     
    14281262where $H_s$ is the significant wave height and $\omega$ is the wave frequency. 
    14291263 
    1430 \item[{\np{nn_sdrift}{nn\_sdrift} = 1}]: velocity profile based on the Phillips spectrum which is considered to be a 
     1264\item [{\np{nn_sdrift}{nn\_sdrift} = 1}]: velocity profile based on the Phillips spectrum which is considered to be a 
    14311265reasonable estimate of the part of the spectrum mostly contributing to the Stokes drift velocity near the surface 
    14321266\citep{breivik.bidlot.ea_OM16}: 
     
    14401274where $erf$ is the complementary error function and $k_p$ is the peak wavenumber. 
    14411275 
    1442 \item[{\np{nn_sdrift}{nn\_sdrift} = 2}]: velocity profile based on the Phillips spectrum as for \np{nn_sdrift}{nn\_sdrift} = 1 
     1276\item [{\np{nn_sdrift}{nn\_sdrift} = 2}]: velocity profile based on the Phillips spectrum as for \np{nn_sdrift}{nn\_sdrift} = 1 
    14431277but using the wave frequency from a wave model. 
    14441278 
     
    14651299  - (\mathbf{U} + \mathbf{U}_{st}) \cdot \nabla{c} 
    14661300\] 
    1467  
    14681301 
    14691302% ---------------------------------------------------------------- 
     
    14791312approximations described in \autoref{subsec:SBC_wave_sdw}), 
    14801313\np[=.true.]{ln_stcor}{ln\_stcor} has to be set. 
    1481  
    14821314 
    14831315% ---------------------------------------------------------------- 
     
    15211353meridional stress components by setting \np[=.true.]{ln_tauw}{ln\_tauw}. 
    15221354 
    1523  
    1524  
    1525 % ================================================================ 
    1526 % Miscellanea options 
    1527 % ================================================================ 
    15281355\section{Miscellaneous options} 
    15291356\label{sec:SBC_misc} 
    15301357 
    1531  
    1532 % ------------------------------------------------------------------------------------------------------------- 
    1533 %        Diurnal cycle 
    1534 % ------------------------------------------------------------------------------------------------------------- 
    15351358\subsection[Diurnal cycle (\textit{sbcdcy.F90})]{Diurnal cycle (\protect\mdl{sbcdcy})} 
    15361359\label{subsec:SBC_dcy} 
     
    15401363%------------------------------------------------------------------------------------------------------------- 
    15411364 
    1542 %>>>>>>>>>>>>>>>>>>>>>>>>>>>> 
    15431365\begin{figure}[!t] 
    15441366  \centering 
     
    15531375  \label{fig:SBC_diurnal} 
    15541376\end{figure} 
    1555 %>>>>>>>>>>>>>>>>>>>>>>>>>>>> 
    15561377 
    15571378\cite{bernie.woolnough.ea_JC05} have shown that to capture 90$\%$ of the diurnal variability of SST requires a vertical resolution in upper ocean of 1~m or better and a temporal resolution of the surface fluxes of 3~h or less. 
     
    15761397one every 2~hours (from 1am to 11pm). 
    15771398 
    1578 %>>>>>>>>>>>>>>>>>>>>>>>>>>>> 
    15791399\begin{figure}[!t] 
    15801400  \centering 
     
    15861406  \label{fig:SBC_dcy} 
    15871407\end{figure} 
    1588 %>>>>>>>>>>>>>>>>>>>>>>>>>>>> 
    15891408 
    15901409Note also that the setting a diurnal cycle in SWF is highly recommended when 
     
    15921411an inconsistency between the scale of the vertical resolution and the forcing acting on that scale. 
    15931412 
    1594  
    1595 % ------------------------------------------------------------------------------------------------------------- 
    1596 %        Rotation of vector pairs onto the model grid directions 
    1597 % ------------------------------------------------------------------------------------------------------------- 
    15981413\subsection{Rotation of vector pairs onto the model grid directions} 
    15991414\label{subsec:SBC_rotation} 
     
    16121427The rot\_rep routine from the \mdl{geo2ocean} module is used to perform the rotation. 
    16131428 
    1614  
    1615 % ------------------------------------------------------------------------------------------------------------- 
    1616 %        Surface restoring to observed SST and/or SSS 
    1617 % ------------------------------------------------------------------------------------------------------------- 
    16181429\subsection[Surface restoring to observed SST and/or SSS (\textit{sbcssr.F90})]{Surface restoring to observed SST and/or SSS (\protect\mdl{sbcssr})} 
    16191430\label{subsec:SBC_ssr} 
     
    16601471reduce the uncertainties we have on the observed freshwater budget. 
    16611472 
    1662  
    1663 % ------------------------------------------------------------------------------------------------------------- 
    1664 %        Handling of ice-covered area 
    1665 % ------------------------------------------------------------------------------------------------------------- 
    16661473\subsection{Handling of ice-covered area  (\textit{sbcice\_...})} 
    16671474\label{subsec:SBC_ice-cover} 
     
    16711478the value of the \np{nn_ice}{nn\_ice} namelist parameter found in \nam{sbc}{sbc} namelist. 
    16721479\begin{description} 
    1673 \item[nn\_ice = 0] 
     1480\item [nn\_ice = 0] 
    16741481  there will never be sea-ice in the computational domain. 
    16751482  This is a typical namelist value used for tropical ocean domain. 
    16761483  The surface fluxes are simply specified for an ice-free ocean. 
    16771484  No specific things is done for sea-ice. 
    1678 \item[nn\_ice = 1] 
     1485\item [nn\_ice = 1] 
    16791486  sea-ice can exist in the computational domain, but no sea-ice model is used. 
    16801487  An observed ice covered area is read in a file. 
     
    16881495  is usually referred as the \textit{ice-if} model. 
    16891496  It can be found in the \mdl{sbcice\_if} module. 
    1690 \item[nn\_ice = 2 or more] 
     1497\item [nn\_ice = 2 or more] 
    16911498  A full sea ice model is used. 
    16921499  This model computes the ice-ocean fluxes, 
     
    17011508%GS: ocean-ice (SI3) interface is not located in SBC directory anymore, so it should be included in SI3 doc 
    17021509 
    1703  
    1704 % ------------------------------------------------------------------------------------------------------------- 
    1705 %        CICE-ocean Interface 
    1706 % ------------------------------------------------------------------------------------------------------------- 
    17071510\subsection[Interface to CICE (\textit{sbcice\_cice.F90})]{Interface to CICE (\protect\mdl{sbcice\_cice})} 
    17081511\label{subsec:SBC_cice} 
     
    17351538there is no sea ice. 
    17361539 
    1737  
    1738 % ------------------------------------------------------------------------------------------------------------- 
    1739 %        Freshwater budget control 
    1740 % ------------------------------------------------------------------------------------------------------------- 
    17411540\subsection[Freshwater budget control (\textit{sbcfwb.F90})]{Freshwater budget control (\protect\mdl{sbcfwb})} 
    17421541\label{subsec:SBC_fwb} 
     
    17471546 
    17481547\begin{description} 
    1749 \item[{\np[=0]{nn_fwb}{nn\_fwb}}] 
     1548\item [{\np[=0]{nn_fwb}{nn\_fwb}}] 
    17501549  no control at all. 
    17511550  The mean sea level is free to drift, and will certainly do so. 
    1752 \item[{\np[=1]{nn_fwb}{nn\_fwb}}] 
     1551\item [{\np[=1]{nn_fwb}{nn\_fwb}}] 
    17531552  global mean \textit{emp} set to zero at each model time step. 
    17541553  %GS: comment below still relevant ? 
    17551554  %Note that with a sea-ice model, this technique only controls the mean sea level with linear free surface and no mass flux between ocean and ice (as it is implemented in the current ice-ocean coupling). 
    1756 \item[{\np[=2]{nn_fwb}{nn\_fwb}}] 
     1555\item [{\np[=2]{nn_fwb}{nn\_fwb}}] 
    17571556  freshwater budget is adjusted from the previous year annual mean budget which 
    17581557  is read in the \textit{EMPave\_old.dat} file. 
  • NEMO/trunk/doc/latex/NEMO/subfiles/chap_STO.tex

    r11584 r11596  
    22 
    33\begin{document} 
    4 % ================================================================ 
    5 % Chapter stochastic parametrization of EOS (STO) 
    6 % ================================================================ 
    74\chapter{Stochastic Parametrization of EOS (STO)} 
    85\label{chap:STO} 
     
    2421P.-A. Bouttier release 3.6 inital version 
    2522 
    26 \newpage 
    27  
    2823As a result of the nonlinearity of the seawater equation of state, unresolved scales represent a major source of uncertainties in the computation of the large-scale horizontal density gradient from the large-scale temperature and salinity fields. Following  \cite{brankart_OM13}, the impact of these uncertainties can be simulated by random processes representing unresolved T/S fluctuations. The Stochastic Parametrization of EOS (STO) module implements this parametrization. 
    2924 
     
    4338a parametrized decorrelation time scale, and horizontal and vertical standard deviations $\sigma_s$. 
    4439$\mathbf{\xi}$ are uncorrelated over the horizontal and fully correlated along the vertical. 
    45  
    4640 
    4741\section{Stochastic processes} 
     
    6963 
    7064\begin{itemize} 
    71 \item 
    72   for order~1 processes, $w^{(i)}$ is a Gaussian white noise, with zero mean and standard deviation equal to~1, 
     65\item for order~1 processes, $w^{(i)}$ is a Gaussian white noise, with zero mean and standard deviation equal to~1, 
    7366  and the parameters $a^{(i)}$, $b^{(i)}$, $c^{(i)}$ are given by: 
    7467 
     
    8477  \] 
    8578 
    86 \item 
    87   for order~$n>1$ processes, $w^{(i)}$ is an order~$n-1$ autoregressive process, with zero mean, 
     79\item for order~$n>1$ processes, $w^{(i)}$ is an order~$n-1$ autoregressive process, with zero mean, 
    8880  standard deviation equal to~$\sigma^{(i)}$; 
    8981  correlation timescale equal to~$\tau^{(i)}$; 
     
    124116\section{Implementation details} 
    125117\label{sec:STO_thech_details} 
    126  
    127118 
    128119The code implementing stochastic parametrisation is located in the src/OCE/STO directory. 
     
    186177 
    187178\begin{description} 
    188 \item[{\np{nn_sto_eos}{nn\_sto\_eos}:}]   number of independent random walks 
    189 \item[{\np{rn_eos_stdxy}{rn\_eos\_stdxy}:}] random walk horizontal standard deviation (in grid points) 
    190 \item[{\np{rn_eos_stdz}{rn\_eos\_stdz}:}]  random walk vertical standard deviation (in grid points) 
    191 \item[{\np{rn_eos_tcor}{rn\_eos\_tcor}:}]  random walk time correlation (in timesteps) 
    192 \item[{\np{nn_eos_ord}{nn\_eos\_ord}:}]   order of autoregressive processes 
    193 \item[{\np{nn_eos_flt}{nn\_eos\_flt}:}]   passes of Laplacian filter 
    194 \item[{\np{rn_eos_lim}{rn\_eos\_lim}:}]   limitation factor (default = 3.0) 
     179\item [{\np{nn_sto_eos}{nn\_sto\_eos}:}]   number of independent random walks 
     180\item [{\np{rn_eos_stdxy}{rn\_eos\_stdxy}:}] random walk horizontal standard deviation (in grid points) 
     181\item [{\np{rn_eos_stdz}{rn\_eos\_stdz}:}]  random walk vertical standard deviation (in grid points) 
     182\item [{\np{rn_eos_tcor}{rn\_eos\_tcor}:}]  random walk time correlation (in timesteps) 
     183\item [{\np{nn_eos_ord}{nn\_eos\_ord}:}]   order of autoregressive processes 
     184\item [{\np{nn_eos_flt}{nn\_eos\_flt}:}]   passes of Laplacian filter 
     185\item [{\np{rn_eos_lim}{rn\_eos\_lim}:}]   limitation factor (default = 3.0) 
    195186\end{description} 
    196187 
  • NEMO/trunk/doc/latex/NEMO/subfiles/chap_TRA.tex

    r11584 r11596  
    22 
    33\begin{document} 
    4 % ================================================================ 
    5 % Chapter 1 ——— Ocean Tracers (TRA) 
    6 % ================================================================ 
    74\chapter{Ocean Tracers (TRA)} 
    85\label{chap:TRA} 
     
    5754(\np{ln_tra_trd}{ln\_tra\_trd} or \np[=.true.]{ln_tra_mxl}{ln\_tra\_mxl}), as described in \autoref{chap:DIA}. 
    5855 
    59 % ================================================================ 
    60 % Tracer Advection 
    61 % ================================================================ 
    6256\section[Tracer advection (\textit{traadv.F90})]{Tracer advection (\protect\mdl{traadv})} 
    6357\label{sec:TRA_adv} 
     
    9185In other words, by setting $\tau = 1$ in (\autoref{eq:TRA_adv}) we recover the discrete form of 
    9286the continuity equation which is used to calculate the vertical velocity. 
    93 %>>>>>>>>>>>>>>>>>>>>>>>>>>>> 
    9487\begin{figure}[!t] 
    9588  \centering 
     
    109102  \label{fig:TRA_adv_scheme} 
    110103\end{figure} 
    111 %>>>>>>>>>>>>>>>>>>>>>>>>>>>> 
    112104 
    113105The key difference between the advection schemes available in \NEMO\ is the choice made in space and 
     
    120112 
    121113\begin{description} 
    122 \item[linear free surface:] 
     114\item [linear free surface:] 
    123115  (\np[=.true.]{ln_linssh}{ln\_linssh}) 
    124116  the first level thickness is constant in time: 
     
    128120  $\tau_w|_{k = 1/2} = T_{k = 1}$, \ie\ the product of surface velocity (at $z = 0$) by 
    129121  the first level tracer value. 
    130 \item[non-linear free surface:] 
     122\item [non-linear free surface:] 
    131123  (\np[=.false.]{ln_linssh}{ln\_linssh}) 
    132124  convergence/divergence in the first ocean level moves the free surface up/down. 
     
    163155 
    164156\begin{enumerate} 
    165 \item 
    166   CEN and FCT schemes require an explicit diffusion operator while the other schemes are diffusive enough so that 
     157\item CEN and FCT schemes require an explicit diffusion operator while the other schemes are diffusive enough so that 
    167158  they do not necessarily need additional diffusion; 
    168 \item 
    169   CEN and UBS are not \textit{positive} schemes 
     159\item CEN and UBS are not \textit{positive} schemes 
    170160  \footnote{negative values can appear in an initially strictly positive tracer field which is advected}, 
    171161  implying that false extrema are permitted. 
    172162  Their use is not recommended on passive tracers; 
    173 \item 
    174   It is recommended that the same advection-diffusion scheme is used on both active and passive tracers. 
     163\item It is recommended that the same advection-diffusion scheme is used on both active and passive tracers. 
    175164\end{enumerate} 
    176165 
     
    182171their results. 
    183172 
    184 % ------------------------------------------------------------------------------------------------------------- 
    185 %        2nd and 4th order centred schemes 
    186 % ------------------------------------------------------------------------------------------------------------- 
    187173\subsection[CEN: Centred scheme (\forcode{ln_traadv_cen})]{CEN: Centred scheme (\protect\np{ln_traadv_cen}{ln\_traadv\_cen})} 
    188174\label{subsec:TRA_adv_cen} 
     
    249235these near boundary grid points. 
    250236 
    251 % ------------------------------------------------------------------------------------------------------------- 
    252 %        FCT scheme 
    253 % ------------------------------------------------------------------------------------------------------------- 
    254237\subsection[FCT: Flux Corrected Transport scheme (\forcode{ln_traadv_fct})]{FCT: Flux Corrected Transport scheme (\protect\np{ln_traadv_fct}{ln\_traadv\_fct})} 
    255238\label{subsec:TRA_adv_tvd} 
     
    285268A comparison of FCT-2 with MUSCL and a MPDATA scheme can be found in \citet{levy.estublier.ea_GRL01}. 
    286269 
    287  
    288270For stability reasons (see \autoref{chap:TD}), 
    289271$\tau_u^{cen}$ is evaluated in (\autoref{eq:TRA_adv_fct}) using the \textit{now} tracer while 
     
    292274while a forward scheme is used for the diffusive part. 
    293275 
    294 % ------------------------------------------------------------------------------------------------------------- 
    295 %        MUSCL scheme 
    296 % ------------------------------------------------------------------------------------------------------------- 
    297276\subsection[MUSCL: Monotone Upstream Scheme for Conservative Laws (\forcode{ln_traadv_mus})]{MUSCL: Monotone Upstream Scheme for Conservative Laws (\protect\np{ln_traadv_mus}{ln\_traadv\_mus})} 
    298277\label{subsec:TRA_adv_mus} 
     
    328307(\np[=.true.]{ln_mus_ups}{ln\_mus\_ups}). 
    329308 
    330 % ------------------------------------------------------------------------------------------------------------- 
    331 %        UBS scheme 
    332 % ------------------------------------------------------------------------------------------------------------- 
    333309\subsection[UBS a.k.a. UP3: Upstream-Biased Scheme (\forcode{ln_traadv_ubs})]{UBS a.k.a. UP3: Upstream-Biased Scheme (\protect\np{ln_traadv_ubs}{ln\_traadv\_ubs})} 
    334310\label{subsec:TRA_adv_ubs} 
     
    400376Note the current version of \NEMO\ uses the computationally more efficient formulation \autoref{eq:TRA_adv_ubs}. 
    401377 
    402 % ------------------------------------------------------------------------------------------------------------- 
    403 %        QCK scheme 
    404 % ------------------------------------------------------------------------------------------------------------- 
    405378\subsection[QCK: QuiCKest scheme (\forcode{ln_traadv_qck})]{QCK: QuiCKest scheme (\protect\np{ln_traadv_qck}{ln\_traadv\_qck})} 
    406379\label{subsec:TRA_adv_qck} 
     
    423396%%%gmcomment   :  Cross term are missing in the current implementation.... 
    424397 
    425 % ================================================================ 
    426 % Tracer Lateral Diffusion 
    427 % ================================================================ 
    428398\section[Tracer lateral diffusion (\textit{traldf.F90})]{Tracer lateral diffusion (\protect\mdl{traldf})} 
    429399\label{sec:TRA_ldf} 
     
    455425the pure vertical component is split into an explicit and an implicit part \citep{lemarie.debreu.ea_OM12}. 
    456426 
    457 % ------------------------------------------------------------------------------------------------------------- 
    458 %        Type of operator 
    459 % ------------------------------------------------------------------------------------------------------------- 
    460427\subsection[Type of operator (\forcode{ln_traldf_}\{\forcode{OFF,lap,blp}\})]{Type of operator (\protect\np{ln_traldf_OFF}{ln\_traldf\_OFF}, \protect\np{ln_traldf_lap}{ln\_traldf\_lap}, or \protect\np{ln_traldf_blp}{ln\_traldf\_blp})} 
    461428\label{subsec:TRA_ldf_op} 
     
    464431 
    465432\begin{description} 
    466 \item[{\np[=.true.]{ln_traldf_OFF}{ln\_traldf\_OFF}}] 
     433\item [{\np[=.true.]{ln_traldf_OFF}{ln\_traldf\_OFF}}] 
    467434  no operator selected, the lateral diffusive tendency will not be applied to the tracer equation. 
    468435  This option can be used when the selected advection scheme is diffusive enough (MUSCL scheme for example). 
    469 \item[{\np[=.true.]{ln_traldf_lap}{ln\_traldf\_lap}}] 
     436\item [{\np[=.true.]{ln_traldf_lap}{ln\_traldf\_lap}}] 
    470437  a laplacian operator is selected. 
    471438  This harmonic operator takes the following expression:  $\mathcal{L}(T) = \nabla \cdot A_{ht} \; \nabla T $, 
    472439  where the gradient operates along the selected direction (see \autoref{subsec:TRA_ldf_dir}), 
    473440  and $A_{ht}$ is the eddy diffusivity coefficient expressed in $m^2/s$ (see \autoref{chap:LDF}). 
    474 \item[{\np[=.true.]{ln_traldf_blp}{ln\_traldf\_blp}}]: 
     441\item [{\np[=.true.]{ln_traldf_blp}{ln\_traldf\_blp}}]: 
    475442  a bilaplacian operator is selected. 
    476443  This biharmonic operator takes the following expression: 
     
    489456whereas the laplacian damping time scales only like $\lambda^{-2}$. 
    490457 
    491 % ------------------------------------------------------------------------------------------------------------- 
    492 %        Direction of action 
    493 % ------------------------------------------------------------------------------------------------------------- 
    494458\subsection[Action direction (\forcode{ln_traldf_}\{\forcode{lev,hor,iso,triad}\})]{Direction of action (\protect\np{ln_traldf_lev}{ln\_traldf\_lev}, \protect\np{ln_traldf_hor}{ln\_traldf\_hor}, \protect\np{ln_traldf_iso}{ln\_traldf\_iso}, or \protect\np{ln_traldf_triad}{ln\_traldf\_triad})} 
    495459\label{subsec:TRA_ldf_dir} 
     
    515479the next two sub-sections. 
    516480 
    517 % ------------------------------------------------------------------------------------------------------------- 
    518 %       iso-level operator 
    519 % ------------------------------------------------------------------------------------------------------------- 
    520481\subsection[Iso-level (bi-)laplacian operator (\forcode{ln_traldf_iso})]{Iso-level (bi-)laplacian operator ( \protect\np{ln_traldf_iso}{ln\_traldf\_iso})} 
    521482\label{subsec:TRA_ldf_lev} 
     
    546507They are calculated in the \mdl{zpshde} module, described in \autoref{sec:TRA_zpshde}. 
    547508 
    548 % ------------------------------------------------------------------------------------------------------------- 
    549 %         Rotated laplacian operator 
    550 % ------------------------------------------------------------------------------------------------------------- 
    551509\subsection{Standard and triad (bi-)laplacian operator} 
    552510\label{subsec:TRA_ldf_iso_triad} 
     
    626584\end{itemize} 
    627585 
    628 % ================================================================ 
    629 % Tracer Vertical Diffusion 
    630 % ================================================================ 
    631586\section[Tracer vertical diffusion (\textit{trazdf.F90})]{Tracer vertical diffusion (\protect\mdl{trazdf})} 
    632587\label{sec:TRA_zdf} 
     
    663618it overcomes the stability constraint. 
    664619 
    665 % ================================================================ 
    666 % External Forcing 
    667 % ================================================================ 
    668620\section{External forcing} 
    669621\label{sec:TRA_sbc_qsr_bbc} 
    670622 
    671 % ------------------------------------------------------------------------------------------------------------- 
    672 %        surface boundary condition 
    673 % ------------------------------------------------------------------------------------------------------------- 
    674623\subsection[Surface boundary condition (\textit{trasbc.F90})]{Surface boundary condition (\protect\mdl{trasbc})} 
    675624\label{subsec:TRA_sbc} 
     
    692641 
    693642\begin{itemize} 
    694 \item 
    695   $Q_{ns}$, the non-solar part of the net surface heat flux that crosses the sea surface 
     643\item $Q_{ns}$, the non-solar part of the net surface heat flux that crosses the sea surface 
    696644  (\ie\ the difference between the total surface heat flux and the fraction of the short wave flux that 
    697645  penetrates into the water column, see \autoref{subsec:TRA_qsr}) 
    698646  plus the heat content associated with of the mass exchange with the atmosphere and lands. 
    699 \item 
    700   $\textit{sfx}$, the salt flux resulting from ice-ocean mass exchange (freezing, melting, ridging...) 
    701 \item 
    702   \textit{emp}, the mass flux exchanged with the atmosphere (evaporation minus precipitation) and 
     647\item $\textit{sfx}$, the salt flux resulting from ice-ocean mass exchange (freezing, melting, ridging...) 
     648\item \textit{emp}, the mass flux exchanged with the atmosphere (evaporation minus precipitation) and 
    703649  possibly with the sea-ice and ice-shelves. 
    704 \item 
    705   \textit{rnf}, the mass flux associated with runoff 
     650\item \textit{rnf}, the mass flux associated with runoff 
    706651  (see \autoref{sec:SBC_rnf} for further detail of how it acts on temperature and salinity tendencies) 
    707 \item 
    708   \textit{fwfisf}, the mass flux associated with ice shelf melt, 
     652\item \textit{fwfisf}, the mass flux associated with ice shelf melt, 
    709653  (see \autoref{sec:SBC_isf} for further details on how the ice shelf melt is computed and applied). 
    710654\end{itemize} 
     
    742686This is the reason why the modified filter is not applied in the linear free surface case (see \autoref{chap:TD}). 
    743687 
    744 % ------------------------------------------------------------------------------------------------------------- 
    745 %        Solar Radiation Penetration 
    746 % ------------------------------------------------------------------------------------------------------------- 
    747688\subsection[Solar radiation penetration (\textit{traqsr.F90})]{Solar radiation penetration (\protect\mdl{traqsr})} 
    748689\label{subsec:TRA_qsr} 
     
    823764 
    824765\begin{description} 
    825 \item[{\np[=0]{nn_chldta}{nn\_chldta}}] 
     766\item [{\np[=0]{nn_chldta}{nn\_chldta}}] 
    826767  a constant 0.05 g.Chl/L value everywhere ; 
    827 \item[{\np[=1]{nn_chldta}{nn\_chldta}}] 
     768\item [{\np[=1]{nn_chldta}{nn\_chldta}}] 
    828769  an observed time varying chlorophyll deduced from satellite surface ocean color measurement spread uniformly in 
    829770  the vertical direction; 
    830 \item[{\np[=2]{nn_chldta}{nn\_chldta}}] 
     771\item [{\np[=2]{nn_chldta}{nn\_chldta}}] 
    831772  same as previous case except that a vertical profile of chlorophyl is used. 
    832773  Following \cite{morel.berthon_LO89}, the profile is computed from the local surface chlorophyll value; 
    833 \item[{\np[=.true.]{ln_qsr_bio}{ln\_qsr\_bio}}] 
     774\item [{\np[=.true.]{ln_qsr_bio}{ln\_qsr\_bio}}] 
    834775  simulated time varying chlorophyll by TOP biogeochemical model. 
    835776  In this case, the RGB formulation is used to calculate both the phytoplankton light limitation in 
     
    849790(\ie\ $I$ is masked). 
    850791 
    851 %>>>>>>>>>>>>>>>>>>>>>>>>>>>> 
    852792\begin{figure}[!t] 
    853793  \centering 
     
    863803  \label{fig:TRA_qsr_irradiance} 
    864804\end{figure} 
    865 %>>>>>>>>>>>>>>>>>>>>>>>>>>>> 
    866  
    867 % ------------------------------------------------------------------------------------------------------------- 
    868 %        Bottom Boundary Condition 
    869 % ------------------------------------------------------------------------------------------------------------- 
     805 
    870806\subsection[Bottom boundary condition (\textit{trabbc.F90}) - \forcode{ln_trabbc})]{Bottom boundary condition (\protect\mdl{trabbc} - \protect\np{ln_trabbc}{ln\_trabbc})} 
    871807\label{subsec:TRA_bbc} 
     
    878814\end{listing} 
    879815%-------------------------------------------------------------------------------------------------------------- 
    880 %>>>>>>>>>>>>>>>>>>>>>>>>>>>> 
    881816\begin{figure}[!t] 
    882817  \centering 
     
    887822  \label{fig:TRA_geothermal} 
    888823\end{figure} 
    889 %>>>>>>>>>>>>>>>>>>>>>>>>>>>> 
    890824 
    891825Usually it is assumed that there is no exchange of heat or salt through the ocean bottom, 
     
    905839the \ifile{geothermal\_heating} NetCDF file (\autoref{fig:TRA_geothermal}) \citep{emile-geay.madec_OS09}. 
    906840 
    907 % ================================================================ 
    908 % Bottom Boundary Layer 
    909 % ================================================================ 
    910841\section[Bottom boundary layer (\textit{trabbl.F90} - \forcode{ln_trabbl})]{Bottom boundary layer (\protect\mdl{trabbl} - \protect\np{ln_trabbl}{ln\_trabbl})} 
    911842\label{sec:TRA_bbl} 
     
    941872\citet{campin.goosse_T99}. 
    942873 
    943 % ------------------------------------------------------------------------------------------------------------- 
    944 %        Diffusive BBL 
    945 % ------------------------------------------------------------------------------------------------------------- 
    946874\subsection[Diffusive bottom boundary layer (\forcode{nn_bbl_ldf=1})]{Diffusive bottom boundary layer (\protect\np[=1]{nn_bbl_ldf}{nn\_bbl\_ldf})} 
    947875\label{subsec:TRA_bbl_diff} 
     
    980908$\overline H^\sigma$, the along bottom mean temperature, salinity and depth, respectively. 
    981909 
    982 % ------------------------------------------------------------------------------------------------------------- 
    983 %        Advective BBL 
    984 % ------------------------------------------------------------------------------------------------------------- 
    985910\subsection[Advective bottom boundary layer (\forcode{nn_bbl_adv=1,2})]{Advective bottom boundary layer (\protect\np[=1,2]{nn_bbl_adv}{nn\_bbl\_adv})} 
    986911\label{subsec:TRA_bbl_adv} 
     
    991916%} 
    992917 
    993 %>>>>>>>>>>>>>>>>>>>>>>>>>>>> 
    994918\begin{figure}[!t] 
    995919  \centering 
     
    1006930  \label{fig:TRA_bbl} 
    1007931\end{figure} 
    1008 %>>>>>>>>>>>>>>>>>>>>>>>>>>>> 
    1009932 
    1010933%!!      nn_bbl_adv = 1   use of the ocean velocity as bbl velocity 
     
    1071994It has to be used to compute the effective velocity as well as the effective overturning circulation. 
    1072995 
    1073 % ================================================================ 
    1074 % Tracer damping 
    1075 % ================================================================ 
    1076996\section[Tracer damping (\textit{tradmp.F90})]{Tracer damping (\protect\mdl{tradmp})} 
    1077997\label{sec:TRA_dmp} 
     
    11301050\path{./tools/DMP_TOOLS}. 
    11311051 
    1132 % ================================================================ 
    1133 % Tracer time evolution 
    1134 % ================================================================ 
    11351052\section[Tracer time evolution (\textit{tranxt.F90})]{Tracer time evolution (\protect\mdl{tranxt})} 
    11361053\label{sec:TRA_nxt} 
     
    11651082$T^{t - \rdt} = T^t$ and $T^t = T_f$. 
    11661083 
    1167 % ================================================================ 
    1168 % Equation of State (eosbn2) 
    1169 % ================================================================ 
    11701084\section[Equation of state (\textit{eosbn2.F90})]{Equation of state (\protect\mdl{eosbn2})} 
    11711085\label{sec:TRA_eosbn2} 
     
    11791093%-------------------------------------------------------------------------------------------------------------- 
    11801094 
    1181 % ------------------------------------------------------------------------------------------------------------- 
    1182 %        Equation of State 
    1183 % ------------------------------------------------------------------------------------------------------------- 
    11841095\subsection[Equation of seawater (\forcode{ln_}\{\forcode{teos10,eos80,seos}\})]{Equation of seawater (\protect\np{ln_teos10}{ln\_teos10}, \protect\np{ln_teos80}{ln\_teos80}, or \protect\np{ln_seos}{ln\_seos})} 
    11851096\label{subsec:TRA_eos} 
    1186  
    11871097 
    11881098The Equation Of Seawater (EOS) is an empirical nonlinear thermodynamic relationship linking seawater density, 
     
    12161126 
    12171127\begin{description} 
    1218 \item[{\np[=.true.]{ln_teos10}{ln\_teos10}}] 
     1128\item [{\np[=.true.]{ln_teos10}{ln\_teos10}}] 
    12191129  the polyTEOS10-bsq equation of seawater \citep{roquet.madec.ea_OM15} is used. 
    12201130  The accuracy of this approximation is comparable to the TEOS-10 rational function approximation, 
     
    12351145  either computing the air-sea and ice-sea fluxes (forced mode) or 
    12361146  sending the SST field to the atmosphere (coupled mode). 
    1237 \item[{\np[=.true.]{ln_eos80}{ln\_eos80}}] 
     1147\item [{\np[=.true.]{ln_eos80}{ln\_eos80}}] 
    12381148  the polyEOS80-bsq equation of seawater is used. 
    12391149  It takes the same polynomial form as the polyTEOS10, but the coefficients have been optimized to 
     
    12471157  Nevertheless, a severe assumption is made in order to have a heat content ($C_p T_p$) which 
    12481158  is conserved by the model: $C_p$ is set to a constant value, the TEOS10 value. 
    1249 \item[{\np[=.true.]{ln_seos}{ln\_seos}}] 
     1159\item [{\np[=.true.]{ln_seos}{ln\_seos}}] 
    12501160  a simplified EOS (S-EOS) inspired by \citet{vallis_bk06} is chosen, 
    12511161  the coefficients of which has been optimized to fit the behavior of TEOS10 
     
    12771187\end{description} 
    12781188 
    1279 %>>>>>>>>>>>>>>>>>>>>>>>>>>>> 
    12801189\begin{table}[!tb] 
    12811190  \centering 
     
    13021211  \label{tab:TRA_SEOS} 
    13031212\end{table} 
    1304 %>>>>>>>>>>>>>>>>>>>>>>>>>>>> 
    1305  
    1306 % ------------------------------------------------------------------------------------------------------------- 
    1307 %        Brunt-V\"{a}is\"{a}l\"{a} Frequency 
    1308 % ------------------------------------------------------------------------------------------------------------- 
     1213 
    13091214\subsection[Brunt-V\"{a}is\"{a}l\"{a} frequency]{Brunt-V\"{a}is\"{a}l\"{a} frequency} 
    13101215\label{subsec:TRA_bn2} 
     
    13271232They are computed through \textit{eos\_rab}, a \fortran\ function that can be found in \mdl{eosbn2}. 
    13281233 
    1329 % ------------------------------------------------------------------------------------------------------------- 
    1330 %        Freezing Point of Seawater 
    1331 % ------------------------------------------------------------------------------------------------------------- 
    13321234\subsection{Freezing point of seawater} 
    13331235\label{subsec:TRA_fzp} 
     
    13491251a \fortran\ function that can be found in \mdl{eosbn2}. 
    13501252 
    1351 % ------------------------------------------------------------------------------------------------------------- 
    1352 %        Potential Energy 
    1353 % ------------------------------------------------------------------------------------------------------------- 
    13541253%\subsection{Potential Energy anomalies} 
    13551254%\label{subsec:TRA_bn2} 
     
    13581257% 
    13591258 
    1360 % ================================================================ 
    1361 % Horizontal Derivative in zps-coordinate 
    1362 % ================================================================ 
    13631259\section[Horizontal derivative in \textit{zps}-coordinate (\textit{zpshde.F90})]{Horizontal derivative in \textit{zps}-coordinate (\protect\mdl{zpshde})} 
    13641260\label{sec:TRA_zpshde} 
     
    13801276For example, for temperature in the $i$-direction the needed interpolated temperature, $\widetilde T$, is: 
    13811277 
    1382 %>>>>>>>>>>>>>>>>>>>>>>>>>>>> 
    13831278\begin{figure}[!p] 
    13841279  \centering 
     
    13981293  \label{fig:TRA_Partial_step_scheme} 
    13991294\end{figure} 
    1400 %>>>>>>>>>>>>>>>>>>>>>>>>>>>> 
    14011295\[ 
    14021296  \widetilde T = \lt\{ 
  • NEMO/trunk/doc/latex/NEMO/subfiles/chap_ZDF.tex

    r11584 r11596  
    55 
    66\begin{document} 
    7 % ================================================================ 
    8 % Chapter  Vertical Ocean Physics (ZDF) 
    9 % ================================================================ 
    107\chapter{Vertical Ocean Physics (ZDF)} 
    118\label{chap:ZDF} 
     
    1512%gm% Add here a small introduction to ZDF and naming of the different physics (similar to what have been written for TRA and DYN. 
    1613 
    17 \newpage 
    18  
    19 % ================================================================ 
    20 % Vertical Mixing 
    21 % ================================================================ 
    2214\section{Vertical mixing} 
    2315\label{sec:ZDF} 
     
    5547%-------------------------------------------------------------------------------------------------------------- 
    5648 
    57 % ------------------------------------------------------------------------------------------------------------- 
    58 %        Constant 
    59 % ------------------------------------------------------------------------------------------------------------- 
    6049\subsection[Constant (\forcode{ln_zdfcst})]{Constant (\protect\np{ln_zdfcst}{ln\_zdfcst})} 
    6150\label{subsec:ZDF_cst} 
     
    7766$\sim10^{-9}~m^2.s^{-1}$ for salinity. 
    7867 
    79 % ------------------------------------------------------------------------------------------------------------- 
    80 %        Richardson Number Dependent 
    81 % ------------------------------------------------------------------------------------------------------------- 
    8268\subsection[Richardson number dependent (\forcode{ln_zdfric})]{Richardson number dependent (\protect\np{ln_zdfric}{ln\_zdfric})} 
    8369\label{subsec:ZDF_ric} 
     
    138124the empirical values \np{rn_wtmix}{rn\_wtmix} and \np{rn_wvmix}{rn\_wvmix} \citep{lermusiaux_JMS01}. 
    139125 
    140 % ------------------------------------------------------------------------------------------------------------- 
    141 %        TKE Turbulent Closure Scheme 
    142 % ------------------------------------------------------------------------------------------------------------- 
    143126\subsection[TKE turbulent closure scheme (\forcode{ln_zdftke})]{TKE turbulent closure scheme (\protect\np{ln_zdftke}{ln\_zdftke})} 
    144127\label{subsec:ZDF_tke} 
     
    248231evaluate the dissipation and mixing length scales as 
    249232(and note that here we use numerical indexing): 
    250 %>>>>>>>>>>>>>>>>>>>>>>>>>>>> 
    251233\begin{figure}[!t] 
    252234  \centering 
     
    255237  \label{fig:ZDF_mixing_length} 
    256238\end{figure} 
    257 %>>>>>>>>>>>>>>>>>>>>>>>>>>>> 
    258239\[ 
    259240  % \label{eq:ZDF_tke_mxl2} 
     
    421402% (\eg\ Mellor, 1989; Large et al., 1994; Meier, 2001; Axell, 2002; St. Laurent and Garrett, 2002). 
    422403 
    423 % ------------------------------------------------------------------------------------------------------------- 
    424 %        GLS Generic Length Scale Scheme 
    425 % ------------------------------------------------------------------------------------------------------------- 
    426404\subsection[GLS: Generic Length Scale (\forcode{ln_zdfgls})]{GLS: Generic Length Scale (\protect\np{ln_zdfgls}{ln\_zdfgls})} 
    427405\label{subsec:ZDF_gls} 
     
    544522 in \citet{reffray.guillaume.ea_GMD15} for the \NEMO\ model. 
    545523 
    546  
    547 % ------------------------------------------------------------------------------------------------------------- 
    548 %        OSM OSMOSIS BL Scheme 
    549 % ------------------------------------------------------------------------------------------------------------- 
    550524\subsection[OSM: OSMosis boundary layer scheme (\forcode{ln_zdfosm})]{OSM: OSMosis boundary layer scheme (\protect\np{ln_zdfosm}{ln\_zdfosm})} 
    551525\label{subsec:ZDF_osm} 
     
    561535The OSMOSIS turbulent closure scheme is based on......   TBC 
    562536 
    563 % ------------------------------------------------------------------------------------------------------------- 
    564 %        TKE and GLS discretization considerations 
    565 % ------------------------------------------------------------------------------------------------------------- 
    566537\subsection[ Discrete energy conservation for TKE and GLS schemes]{Discrete energy conservation for TKE and GLS schemes} 
    567538\label{subsec:ZDF_tke_ene} 
    568539 
    569 %>>>>>>>>>>>>>>>>>>>>>>>>>>>> 
    570540\begin{figure}[!t] 
    571541  \centering 
     
    576546  \label{fig:ZDF_TKE_time_scheme} 
    577547\end{figure} 
    578 %>>>>>>>>>>>>>>>>>>>>>>>>>>>> 
    579548 
    580549The production of turbulence by vertical shear (the first term of the right hand side of 
     
    666635%For the latter, it is in fact the ratio $\sqrt{\bar{e}}/l_\epsilon$ which is stored. 
    667636 
    668 % ================================================================ 
    669 % Convection 
    670 % ================================================================ 
    671637\section{Convection} 
    672638\label{sec:ZDF_conv} 
     
    679645or/and the use of a turbulent closure scheme. 
    680646 
    681 % ------------------------------------------------------------------------------------------------------------- 
    682 %       Non-Penetrative Convective Adjustment 
    683 % ------------------------------------------------------------------------------------------------------------- 
    684647\subsection[Non-penetrative convective adjustment (\forcode{ln_tranpc})]{Non-penetrative convective adjustment (\protect\np{ln_tranpc}{ln\_tranpc})} 
    685648\label{subsec:ZDF_npc} 
    686649 
    687 %>>>>>>>>>>>>>>>>>>>>>>>>>>>> 
    688650\begin{figure}[!htb] 
    689651  \centering 
     
    704666  \label{fig:ZDF_npc} 
    705667\end{figure} 
    706 %>>>>>>>>>>>>>>>>>>>>>>>>>>>> 
    707668 
    708669Options are defined through the \nam{zdf}{zdf} namelist variables. 
     
    744705having to recompute the expansion coefficients at each mixing iteration. 
    745706 
    746 % ------------------------------------------------------------------------------------------------------------- 
    747 %       Enhanced Vertical Diffusion 
    748 % ------------------------------------------------------------------------------------------------------------- 
    749707\subsection[Enhanced vertical diffusion (\forcode{ln_zdfevd})]{Enhanced vertical diffusion (\protect\np{ln_zdfevd}{ln\_zdfevd})} 
    750708\label{subsec:ZDF_evd} 
     
    770728a leapfrog environment \citep{leclair_phd10} (see \autoref{sec:TD_mLF}). 
    771729 
    772 % ------------------------------------------------------------------------------------------------------------- 
    773 %       Turbulent Closure Scheme 
    774 % ------------------------------------------------------------------------------------------------------------- 
    775730\subsection[Handling convection with turbulent closure schemes (\forcode{ln_zdf_}\{\forcode{tke,gls,osm}\})]{Handling convection with turbulent closure schemes (\forcode{ln_zdf{tke,gls,osm}})} 
    776731\label{subsec:ZDF_tcs} 
    777  
    778732 
    779733The turbulent closure schemes presented in \autoref{subsec:ZDF_tke}, \autoref{subsec:ZDF_gls} and 
     
    798752% gm%  + one word on non local flux with KPP scheme trakpp.F90 module... 
    799753 
    800 % ================================================================ 
    801 % Double Diffusion Mixing 
    802 % ================================================================ 
    803754\section[Double diffusion mixing (\forcode{ln_zdfddm})]{Double diffusion mixing (\protect\np{ln_zdfddm}{ln\_zdfddm})} 
    804755\label{subsec:ZDF_ddm} 
    805  
    806756 
    807757%-------------------------------------------namzdf_ddm------------------------------------------------- 
     
    818768it leads to relatively minor changes in circulation but exerts significant regional influences on 
    819769temperature and salinity. 
    820  
    821770 
    822771Diapycnal mixing of S and T are described by diapycnal diffusion coefficients 
     
    844793\end{align} 
    845794 
    846 %>>>>>>>>>>>>>>>>>>>>>>>>>>>> 
    847795\begin{figure}[!t] 
    848796  \centering 
     
    861809  \label{fig:ZDF_ddm} 
    862810\end{figure} 
    863 %>>>>>>>>>>>>>>>>>>>>>>>>>>>> 
    864811 
    865812The factor 0.7 in \autoref{eq:ZDF_ddm_f_T} reflects the measured ratio $\alpha F_T /\beta F_S \approx  0.7$ of 
     
    893840This avoids duplication in the computation of $\alpha$ and $\beta$ (which is usually quite expensive). 
    894841 
    895 % ================================================================ 
    896 % Bottom Friction 
    897 % ================================================================ 
    898842\section[Bottom and top friction (\textit{zdfdrg.F90})]{Bottom and top friction (\protect\mdl{zdfdrg})} 
    899843\label{sec:ZDF_drg} 
     
    924868As the friction processes at the top and the bottom are treated in and identical way, 
    925869the description below considers mostly the bottom friction case, if not stated otherwise. 
    926  
    927870 
    928871Both the surface momentum flux (wind stress) and the bottom momentum flux (bottom friction) enter the equations as 
     
    973916Note than from \NEMO\ 4.0, drag coefficients are only computed at cell centers (\ie\ at T-points) and refer to as $c_b^T$ in the following. These are then linearly interpolated in space to get $c_b^\textbf{U}$ at velocity points. 
    974917 
    975 % ------------------------------------------------------------------------------------------------------------- 
    976 %       Linear Bottom Friction 
    977 % ------------------------------------------------------------------------------------------------------------- 
    978918\subsection[Linear top/bottom friction (\forcode{ln_lin})]{Linear top/bottom friction (\protect\np{ln_lin}{ln\_lin})} 
    979919\label{subsec:ZDF_drg_linear} 
     
    1012952$mask\_value$ * \np{rn_boost}{rn\_boost} * \np{rn_Cd0}{rn\_Cd0}. 
    1013953 
    1014 % ------------------------------------------------------------------------------------------------------------- 
    1015 %       Non-Linear Bottom Friction 
    1016 % ------------------------------------------------------------------------------------------------------------- 
    1017954\subsection[Non-linear top/bottom friction (\forcode{ln_non_lin})]{Non-linear top/bottom friction (\protect\np{ln_non_lin}{ln\_non\_lin})} 
    1018955\label{subsec:ZDF_drg_nonlinear} 
     
    1047984$mask\_value$ * \np{rn_boost}{rn\_boost} * \np{rn_Cd0}{rn\_Cd0}. 
    1048985 
    1049 % ------------------------------------------------------------------------------------------------------------- 
    1050 %       Bottom Friction Log-layer 
    1051 % ------------------------------------------------------------------------------------------------------------- 
    1052986\subsection[Log-layer top/bottom friction (\forcode{ln_loglayer})]{Log-layer top/bottom friction (\protect\np{ln_loglayer}{ln\_loglayer})} 
    1053987\label{subsec:ZDF_drg_loglayer} 
     
    10731007%In this case, the relevant namelist parameters are \np{rn_tfrz0}{rn\_tfrz0}, \np{rn_tfri2}{rn\_tfri2} and \np{rn_tfri2_max}{rn\_tfri2\_max}. 
    10741008 
    1075 % ------------------------------------------------------------------------------------------------------------- 
    1076 %       Explicit bottom Friction 
    1077 % ------------------------------------------------------------------------------------------------------------- 
    10781009\subsection[Explicit top/bottom friction (\forcode{ln_drgimp=.false.})]{Explicit top/bottom friction (\protect\np[=.false.]{ln_drgimp}{ln\_drgimp})} 
    10791010\label{subsec:ZDF_drg_stability} 
     
    11341065The number of potential breaches of the explicit stability criterion are still reported for information purposes. 
    11351066 
    1136 % ------------------------------------------------------------------------------------------------------------- 
    1137 %       Implicit Bottom Friction 
    1138 % ------------------------------------------------------------------------------------------------------------- 
    11391067\subsection[Implicit top/bottom friction (\forcode{ln_drgimp=.true.})]{Implicit top/bottom friction (\protect\np[=.true.]{ln_drgimp}{ln\_drgimp})} 
    11401068\label{subsec:ZDF_drg_imp} 
     
    11641092Superscript $n+1$ means the velocity used in the friction formula is to be calculated, so it is implicit. 
    11651093 
    1166 % ------------------------------------------------------------------------------------------------------------- 
    1167 %       Bottom Friction with split-explicit free surface 
    1168 % ------------------------------------------------------------------------------------------------------------- 
    11691094\subsection[Bottom friction with split-explicit free surface]{Bottom friction with split-explicit free surface} 
    11701095\label{subsec:ZDF_drg_ts} 
     
    11801105Note that other strategies are possible, like considering vertical diffusion step in advance, \ie\ prior barotropic integration. 
    11811106 
    1182  
    1183 % ================================================================ 
    1184 % Internal wave-driven mixing 
    1185 % ================================================================ 
    11861107\section[Internal wave-driven mixing (\forcode{ln_zdfiwm})]{Internal wave-driven mixing (\protect\np{ln_zdfiwm}{ln\_zdfiwm})} 
    11871108\label{subsec:ZDF_tmx_new} 
     
    12451166% Jc: input files names ? 
    12461167 
    1247 % ================================================================ 
    1248 % surface wave-induced mixing 
    1249 % ================================================================ 
    12501168\section[Surface wave-induced mixing (\forcode{ln_zdfswm})]{Surface wave-induced mixing (\protect\np{ln_zdfswm}{ln\_zdfswm})} 
    12511169\label{subsec:ZDF_swm} 
     
    12781196(for more information on wave parameters and settings see \autoref{sec:SBC_wave}) 
    12791197 
    1280 % ================================================================ 
    1281 % Adaptive-implicit vertical advection 
    1282 % ================================================================ 
    12831198\section[Adaptive-implicit vertical advection (\forcode{ln_zad_Aimp})]{Adaptive-implicit vertical advection(\protect\np{ln_zad_Aimp}{ln\_zad\_Aimp})} 
    12841199\label{subsec:ZDF_aimp} 
  • NEMO/trunk/doc/latex/NEMO/subfiles/chap_cfgs.tex

    r11584 r11596  
    22 
    33\begin{document} 
    4 % ================================================================ 
    5 % Chapter Configurations 
    6 % ================================================================ 
    74\chapter{Configurations} 
    85\label{chap:CFGS} 
     
    107\chaptertoc 
    118 
    12 \newpage 
    13  
    14 % ================================================================ 
    15 % Introduction 
    16 % ================================================================ 
    179\section{Introduction} 
    1810\label{sec:CFGS_intro} 
     
    3527%------------------------------------------------------------------------------------------------------------- 
    3628 
    37 % ================================================================ 
    38 % 1D model configuration 
    39 % ================================================================ 
    4029\section[C1D: 1D Water column model (\texttt{\textbf{key\_c1d}})]{C1D: 1D Water column model (\protect\key{c1d})} 
    4130\label{sec:CFGS_c1d} 
     
    6049Therefore, defining \key{c1d} changes some things in the code behaviour: 
    6150\begin{description} 
    62 \item[(1)] 
     51\item [(1)] 
    6352  a simplified \rou{stp} routine is used (\rou{stp\_c1d}, see \mdl{step\_c1d} module) in which 
    6453  both lateral tendancy terms and lateral physics are not called; 
    65 \item[(2)] 
     54\item [(2)] 
    6655  the vertical velocity is zero 
    6756  (so far, no attempt at introducing a Ekman pumping velocity has been made); 
    68 \item[(3)] 
     57\item [(3)] 
    6958  a simplified treatment of the Coriolis term is performed as $U$- and $V$-points are the same 
    7059  (see \mdl{dyncor\_c1d}). 
     
    7564% to be added:  a test case on the yearlong Ocean Weather Station (OWS) Papa dataset of Martin (1985) 
    7665 
    77 % ================================================================ 
    78 % ORCA family configurations 
    79 % ================================================================ 
    8066\section{ORCA family: global ocean with tripolar grid} 
    8167\label{sec:CFGS_orca} 
     
    9076In this namelist\_cfg the name of domain input file is set in \nam{cfg}{cfg} block of namelist. 
    9177 
    92 %>>>>>>>>>>>>>>>>>>>>>>>>>>>> 
    9378\begin{figure}[!t] 
    9479  \centering 
     
    10489  \label{fig:CFGS_ORCA_msh} 
    10590\end{figure} 
    106 %>>>>>>>>>>>>>>>>>>>>>>>>>>>> 
    107  
    108 % ------------------------------------------------------------------------------------------------------------- 
    109 %       ORCA tripolar grid 
    110 % ------------------------------------------------------------------------------------------------------------- 
     91 
    11192\subsection{ORCA tripolar grid} 
    11293\label{subsec:CFGS_orca_grid} 
     
    121102The resulting mesh presents no loss of continuity in either the mesh lines or the scale factors, 
    122103or even the scale factor derivatives over the whole ocean domain, as the mesh is not a composite mesh. 
    123 %>>>>>>>>>>>>>>>>>>>>>>>>>>>> 
    124104\begin{figure}[!tbp] 
    125105  \centering 
     
    136116  \label{fig:CFGS_ORCA_e1e2} 
    137117\end{figure} 
    138 %>>>>>>>>>>>>>>>>>>>>>>>>>>>> 
    139118 
    140119The method is applied to Mercator grid (\ie\ same zonal and meridional grid spacing) poleward of 20\deg{N}, 
     
    149128while the ratio of anisotropy remains close to one except near the Victoria Island in the Canadian Archipelago. 
    150129 
    151 % ------------------------------------------------------------------------------------------------------------- 
    152 %       ORCA-ICE(-PISCES) configurations 
    153 % ------------------------------------------------------------------------------------------------------------- 
    154130\subsection{ORCA pre-defined resolution} 
    155131\label{subsec:CFGS_orca_resolution} 
     
    182158%-------------------------------------------------------------------------------------------------------------- 
    183159 
    184  
    185160The ORCA\_R2 configuration has the following specificity: starting from a 2\deg\ ORCA mesh, 
    186161local mesh refinements were applied to the Mediterranean, Red, Black and Caspian Seas, 
     
    221196sponge layers at open boundaries. 
    222197 
    223 % ------------------------------------------------------------------------------------------------------------- 
    224 %       GYRE family: double gyre basin 
    225 % ------------------------------------------------------------------------------------------------------------- 
    226198\section{GYRE family: double gyre basin} 
    227199\label{sec:CFGS_gyre} 
     
    274246namelist \path{./cfgs/GYRE_PISCES/EXPREF/namelist_cfg}. 
    275247 
    276 %>>>>>>>>>>>>>>>>>>>>>>>>>>>> 
    277248\begin{figure}[!t] 
    278249  \centering 
     
    283254  \label{fig:CFGS_GYRE} 
    284255\end{figure} 
    285 %>>>>>>>>>>>>>>>>>>>>>>>>>>>> 
    286  
    287 % ------------------------------------------------------------------------------------------------------------- 
    288 %       AMM configuration 
    289 % ------------------------------------------------------------------------------------------------------------- 
     256 
    290257\section{AMM: atlantic margin configuration} 
    291258\label{sec:CFGS_config_AMM} 
  • NEMO/trunk/doc/latex/NEMO/subfiles/chap_conservation.tex

    r11584 r11596  
    33\begin{document} 
    44 
    5 % ================================================================ 
    6 % Invariant of the Equations 
    7 % ================================================================ 
    85\chapter{Invariants of the Primitive Equations} 
    96\label{chap:CONS} 
     
    4239\citep{Marti1992?, Levy1996?, Levy1998?}. 
    4340 
    44 % ------------------------------------------------------------------------------------------------------------- 
    45 %       Conservation Properties on Ocean Dynamics 
    46 % ------------------------------------------------------------------------------------------------------------- 
    4741\section{Conservation properties on ocean dynamics} 
    4842\label{sec:CONS_Invariant_dyn} 
     
    158152otherwise there is no guarantee that the surface pressure force work vanishes. 
    159153 
    160 % ------------------------------------------------------------------------------------------------------------- 
    161 %       Conservation Properties on Ocean Thermodynamics 
    162 % ------------------------------------------------------------------------------------------------------------- 
    163154\section{Conservation properties on ocean thermodynamics} 
    164155\label{sec:CONS_Invariant_tra} 
     
    179170In practice, the mass is conserved with a very good accuracy. 
    180171 
    181 % ------------------------------------------------------------------------------------------------------------- 
    182 %       Conservation Properties on Momentum Physics 
    183 % ------------------------------------------------------------------------------------------------------------- 
    184172\subsection{Conservation properties on momentum physics} 
    185173\label{subsec:CONS_Invariant_dyn_physics} 
     
    223211          {A^{lm}\;\zeta \;{\mathrm {\mathbf k}}} \right)} \right]\;dv} \leqslant 0 
    224212\] 
    225  
    226213 
    227214(II.4.6a) and (II.4.6b) means that the horizontal diffusion of momentum conserve both the potential vorticity and 
     
    286273\ie\ the vertical momentum physics conserve momentum, potential vorticity, and horizontal divergence. 
    287274 
    288 % ------------------------------------------------------------------------------------------------------------- 
    289 %       Conservation Properties on Tracer Physics 
    290 % ------------------------------------------------------------------------------------------------------------- 
    291275\subsection{Conservation properties on tracer physics} 
    292276\label{subsec:CONS_Invariant_tra_physics} 
  • NEMO/trunk/doc/latex/NEMO/subfiles/chap_misc.tex

    r11584 r11596  
    22 
    33\begin{document} 
    4 % ================================================================ 
    5 % Chapter --- Miscellaneous Topics 
    6 % ================================================================ 
    74\chapter{Miscellaneous Topics} 
    85\label{chap:MISC} 
     
    107\chaptertoc 
    118 
    12 \newpage 
    13  
    14 % ================================================================ 
    15 % Representation of Unresolved Straits 
    16 % ================================================================ 
    179\section{Representation of unresolved straits} 
    1810\label{sec:MISC_strait} 
     
    3628lateral friction. 
    3729 
    38 % ------------------------------------------------------------------------------------------------------------- 
    39 %       Hand made geometry changes 
    40 % ------------------------------------------------------------------------------------------------------------- 
    4130\subsection{Hand made geometry changes} 
    4231\label{subsec:MISC_strait_hand} 
     
    8372\texttt{fmask} for any other configuration. 
    8473 
    85 %>>>>>>>>>>>>>>>>>>>>>>>>>>>> 
    8674\begin{figure}[!tbp] 
    8775  \centering 
     
    10088  \label{fig:MISC_strait_hand} 
    10189\end{figure} 
    102 %>>>>>>>>>>>>>>>>>>>>>>>>>>>> 
    103  
    104 %>>>>>>>>>>>>>>>>>>>>>>>>>>>> 
     90 
    10591\begin{figure}[!tbp] 
    10692  \centering 
     
    116102  \label{fig:MISC_closea_mask_example} 
    117103\end{figure} 
    118 %>>>>>>>>>>>>>>>>>>>>>>>>>>>> 
    119  
    120 % ================================================================ 
    121 % Closed seas 
    122 % ================================================================ 
     104 
    123105\section[Closed seas (\textit{closea.F90})]{Closed seas (\protect\mdl{closea})} 
    124106\label{sec:MISC_closea} 
     
    141123 
    142124\begin{enumerate} 
    143 \item{{\bfseries No ``closea\_mask'' field is included in domain configuration 
     125\item {{\bfseries No ``closea\_mask'' field is included in domain configuration 
    144126  file.} In this case the closea module does nothing.} 
    145127 
    146 \item{{\bfseries A field called closea\_mask is included in the domain 
     128\item {{\bfseries A field called closea\_mask is included in the domain 
    147129configuration file and ln\_closea=.false. in namelist namcfg.} In this 
    148130case the inland seas defined by the closea\_mask field are filled in 
     
    150132closea\_mask that is nonzero is set to be a land point.} 
    151133 
    152 \item{{\bfseries A field called closea\_mask is included in the domain 
     134\item {{\bfseries A field called closea\_mask is included in the domain 
    153135configuration file and ln\_closea=.true. in namelist namcfg.} Each 
    154136inland sea or group of inland seas is set to a positive integer value 
     
    159141closea\_mask is zero).} 
    160142 
    161 \item{{\bfseries Fields called closea\_mask and closea\_mask\_rnf are 
     143\item {{\bfseries Fields called closea\_mask and closea\_mask\_rnf are 
    162144included in the domain configuration file and ln\_closea=.true. in 
    163145namelist namcfg.} This option works as for option 3, except that if 
     
    173155ocean.} 
    174156 
    175 \item{{\bfseries Fields called closea\_mask and closea\_mask\_emp are 
     157\item {{\bfseries Fields called closea\_mask and closea\_mask\_emp are 
    176158included in the domain configuration file and ln\_closea=.true. in 
    177159namelist namcfg.} This option works the same as option 4 except that 
     
    185167them to the domain configuration file in the utils/tools/DOMAINcfg directory. 
    186168 
    187 % ================================================================ 
    188 % Sub-Domain Functionality 
    189 % ================================================================ 
    190169\section{Sub-domain functionality} 
    191170\label{sec:MISC_zoom} 
     
    213192 
    214193\begin{itemize} 
    215 \item  Add the new attribute to any input files requiring a j-row offset, i.e: 
     194\item Add the new attribute to any input files requiring a j-row offset, i.e: 
    216195\begin{cmds} 
    217196ncatted  -a open_ocean_jstart,global,a,d,41 eORCA1_domcfg.nc 
     
    251230conditions. Experimenting with this remains an exercise for the user. 
    252231 
    253 % ================================================================ 
    254 % Accuracy and Reproducibility 
    255 % ================================================================ 
    256232\section[Accuracy and reproducibility (\textit{lib\_fortran.F90})]{Accuracy and reproducibility (\protect\mdl{lib\_fortran})} 
    257233\label{sec:MISC_fortran} 
     
    281257We use a CPP key as the overwritting of a intrinsic function can present performance issues with 
    282258some computers/compilers. 
    283  
    284259 
    285260\subsection{MPP reproducibility} 
     
    336311non-reference configuration. 
    337312 
    338 % ================================================================ 
    339 % Model optimisation, Control Print and Benchmark 
    340 % ================================================================ 
    341313\section{Model optimisation, control print and benchmark} 
    342314\label{sec:MISC_opt} 
     
    367339 
    368340\begin{enumerate} 
    369 \item{\np{ln_ctl}{ln\_ctl}: compute and print the trends averaged over the interior domain in all TRA, DYN, LDF and 
     341\item {\np{ln_ctl}{ln\_ctl}: compute and print the trends averaged over the interior domain in all TRA, DYN, LDF and 
    370342ZDF modules. 
    371343This option is very helpful when diagnosing the origin of an undesired change in model results. } 
    372344 
    373 \item{also \np{ln_ctl}{ln\_ctl} but using the nictl and njctl namelist parameters to check the source of differences between 
     345\item {also \np{ln_ctl}{ln\_ctl} but using the nictl and njctl namelist parameters to check the source of differences between 
    374346mono and multi processor runs.} 
    375347\end{enumerate} 
     
    415387increment also applies to the time.step file which is otherwise updated every timestep. 
    416388 
    417 % ================================================================ 
    418  
    419 \onlyinsubfile{\input{../../global/epilogue}} 
    420  
    421 \end{document} 
  • NEMO/trunk/doc/latex/NEMO/subfiles/chap_model_basics.tex

    r11584 r11596  
    44\begin{document} 
    55 
    6 % ================================================================ 
    7 % Chapter 1  Model Basics 
    8 % ================================================================ 
    96\chapter{Model Basics} 
    107\label{chap:MB} 
     
    129\chaptertoc 
    1310 
    14 \newpage 
    15  
    16 % ================================================================ 
    17 % Primitive Equations 
    18 % ================================================================ 
     11%% ================================================================================================= 
    1912\section{Primitive equations} 
    2013\label{sec:MB_PE} 
    2114 
    22 % ------------------------------------------------------------------------------------------------------------- 
    23 %        Vector Invariant Formulation 
    24 % ------------------------------------------------------------------------------------------------------------- 
    25  
     15%% ================================================================================================= 
    2616\subsection{Vector invariant formulation} 
    2717\label{subsec:MB_PE_vector} 
     
    3323 
    3424\begin{enumerate} 
    35 \item 
    36   \textit{spherical Earth approximation}: the geopotential surfaces are assumed to be oblate spheriods 
     25\item \textit{spherical Earth approximation}: the geopotential surfaces are assumed to be oblate spheriods 
    3726  that follow the Earth's bulge; these spheroids are approximated by spheres with 
    3827  gravity locally vertical (parallel to the Earth's radius) and independent of latitude 
    3928  \citep[][section 2]{white.hoskins.ea_QJRMS05}. 
    40 \item 
    41   \textit{thin-shell approximation}: the ocean depth is neglected compared to the earth's radius 
    42 \item 
    43   \textit{turbulent closure hypothesis}: the turbulent fluxes 
     29\item \textit{thin-shell approximation}: the ocean depth is neglected compared to the earth's radius 
     30\item \textit{turbulent closure hypothesis}: the turbulent fluxes 
    4431  (which represent the effect of small scale processes on the large-scale) 
    4532  are expressed in terms of large-scale features 
    46 \item 
    47   \textit{Boussinesq hypothesis}: density variations are neglected except in their contribution to 
     33\item \textit{Boussinesq hypothesis}: density variations are neglected except in their contribution to 
    4834  the buoyancy force 
    4935  \begin{equation} 
     
    5137    \rho = \rho \ (T,S,p) 
    5238  \end{equation} 
    53 \item 
    54   \textit{Hydrostatic hypothesis}: the vertical momentum equation is reduced to a balance between 
     39\item \textit{Hydrostatic hypothesis}: the vertical momentum equation is reduced to a balance between 
    5540  the vertical pressure gradient and the buoyancy force 
    5641  (this removes convective processes from the initial Navier-Stokes equations and so 
     
    6045    \pd[p]{z} = - \rho \ g 
    6146  \end{equation} 
    62 \item 
    63   \textit{Incompressibility hypothesis}: the three dimensional divergence of the velocity vector $\vect U$ 
     47\item \textit{Incompressibility hypothesis}: the three dimensional divergence of the velocity vector $\vect U$ 
    6448  is assumed to be zero. 
    6549  \begin{equation} 
     
    6751    \nabla \cdot \vect U = 0 
    6852  \end{equation} 
    69  \item 
    70   \textit{Neglect of additional Coriolis terms}: the Coriolis terms that vary with the cosine of latitude are neglected. 
     53\item \textit{Neglect of additional Coriolis terms}: the Coriolis terms that vary with the cosine of latitude are neglected. 
    7154  These terms may be non-negligible where the Brunt-Vaisala frequency $N$ is small, either in the deep ocean or 
    7255  in the sub-mesoscale motions of the mixed layer, or near the equator \citep[][section 1]{white.hoskins.ea_QJRMS05}. 
     
    10891Their nature and formulation are discussed in \autoref{sec:MB_zdf_ldf} and \autoref{subsec:MB_boundary_condition}. 
    10992 
    110 % ------------------------------------------------------------------------------------------------------------- 
    111 % Boundary condition 
    112 % ------------------------------------------------------------------------------------------------------------- 
     93%% ================================================================================================= 
    11394\subsection{Boundary conditions} 
    11495\label{subsec:MB_boundary_condition} 
     
    128109the other components of the earth system. 
    129110 
    130 %>>>>>>>>>>>>>>>>>>>>>>>>>>>> 
    131111\begin{figure}[!ht] 
    132112  \centering 
     
    138118  \label{fig:MB_ocean_bc} 
    139119\end{figure} 
    140 %>>>>>>>>>>>>>>>>>>>>>>>>>>>> 
    141120 
    142121\begin{description} 
    143 \item[Land - ocean interface:] 
    144   the major flux between continental margins and the ocean is a mass exchange of fresh water through river runoff. 
     122\item [Land - ocean interface:]  the major flux between continental margins and the ocean is a mass exchange of fresh water through river runoff. 
    145123  Such an exchange modifies the sea surface salinity especially in the vicinity of major river mouths. 
    146124  It can be neglected for short range integrations but has to be taken into account for long term integrations as 
     
    148126  It is required in order to close the water cycle of the climate system. 
    149127  It is usually specified as a fresh water flux at the air-sea interface in the vicinity of river mouths. 
    150 \item[Solid earth - ocean interface:] 
    151   heat and salt fluxes through the sea floor are small, except in special areas of little extent. 
     128\item [Solid earth - ocean interface:]  heat and salt fluxes through the sea floor are small, except in special areas of little extent. 
    152129  They are usually neglected in the model 
    153130  \footnote{ 
     
    171148  $\vect D^{\vect U}$ in \autoref{eq:MB_PE_dyn}. 
    172149  It is discussed in \autoref{eq:MB_zdf}.% and Chap. III.6 to 9. 
    173 \item[Atmosphere - ocean interface:] 
    174   the kinematic surface condition plus the mass flux of fresh water PE (the precipitation minus evaporation budget) 
     150\item [Atmosphere - ocean interface:]  the kinematic surface condition plus the mass flux of fresh water PE (the precipitation minus evaporation budget) 
    175151  leads to: 
    176152  \[ 
     
    181157  leads to the continuity of pressure across the interface $z = \eta$. 
    182158  The atmosphere and ocean also exchange horizontal momentum (wind stress), and heat. 
    183 \item[Sea ice - ocean interface:] 
    184   the ocean and sea ice exchange heat, salt, fresh water and momentum. 
     159\item [Sea ice - ocean interface:]  the ocean and sea ice exchange heat, salt, fresh water and momentum. 
    185160  The sea surface temperature is constrained to be at the freezing point at the interface. 
    186161  Sea ice salinity is very low ($\sim4-6 \, psu$) compared to those of the ocean ($\sim34 \, psu$). 
     
    188163\end{description} 
    189164 
    190 % ================================================================ 
    191 % The Horizontal Pressure Gradient 
    192 % ================================================================ 
     165%% ================================================================================================= 
    193166\section{Horizontal pressure gradient} 
    194167\label{sec:MB_hor_pg} 
    195168 
    196 % ------------------------------------------------------------------------------------------------------------- 
    197 % Pressure Formulation 
    198 % ------------------------------------------------------------------------------------------------------------- 
     169%% ================================================================================================= 
    199170\subsection{Pressure formulation} 
    200171\label{subsec:MB_p_formulation} 
     
    228199Only the free surface formulation is now described in this document (see the next sub-section). 
    229200 
    230 % ------------------------------------------------------------------------------------------------------------- 
    231 % Free Surface Formulation 
    232 % ------------------------------------------------------------------------------------------------------------- 
     201%% ================================================================================================= 
    233202\subsection{Free surface formulation} 
    234203\label{subsec:MB_free_surface} 
     
    280249(see \autoref{subsec:DYN_spg_ts}). 
    281250 
    282 % ================================================================ 
    283 % Curvilinear z-coordinate System 
    284 % ================================================================ 
     251%% ================================================================================================= 
    285252\section{Curvilinear \textit{z-}coordinate system} 
    286253\label{sec:MB_zco} 
    287254 
    288 % ------------------------------------------------------------------------------------------------------------- 
    289 % Tensorial Formalism 
    290 % ------------------------------------------------------------------------------------------------------------- 
     255%% ================================================================================================= 
    291256\subsection{Tensorial formalism} 
    292257\label{subsec:MB_tensorial} 
     
    338303  \label{fig:MB_referential} 
    339304\end{figure} 
    340 %>>>>>>>>>>>>>>>>>>>>>>>>>>>> 
    341305 
    342306Since the ocean depth is far smaller than the earth's radius, $a + z$, can be replaced by $a$ in 
     
    373337where $q$ is a scalar quantity and $\vect A = (a_1,a_2,a_3)$ a vector in the $(i,j,k)$ coordinates system. 
    374338 
    375 % ------------------------------------------------------------------------------------------------------------- 
    376 % Continuous Model Equations 
    377 % ------------------------------------------------------------------------------------------------------------- 
     339%% ================================================================================================= 
    378340\subsection{Continuous model equations} 
    379341\label{subsec:MB_zco_Eq} 
     
    496458 
    497459\begin{itemize} 
    498 \item 
    499   \textbf{Vector invariant form of the momentum equations}: 
     460\item \textbf{Vector invariant form of the momentum equations}: 
    500461  \begin{equation} 
    501462    \label{eq:MB_dyn_vect} 
     
    510471    \end{split} 
    511472  \end{equation} 
    512 \item 
    513   \textbf{flux form of the momentum equations}: 
     473\item \textbf{flux form of the momentum equations}: 
    514474  % \label{eq:MB_dyn_flux} 
    515475  \begin{multline*} 
     
    544504  where the divergence of the horizontal velocity, $\chi$ is given by \autoref{eq:MB_div_Uh}. 
    545505 
    546 \item 
    547   \textbf{tracer equations}: 
     506\item \textbf{tracer equations}: 
    548507  \begin{equation} 
    549508  \begin{split} 
     
    562521are discussed in \autoref{chap:SBC}. 
    563522 
    564 \newpage 
    565  
    566 % ================================================================ 
    567 % Curvilinear generalised vertical coordinate System 
    568 % ================================================================ 
     523%% ================================================================================================= 
    569524\section{Curvilinear generalised vertical coordinate system} 
    570525\label{sec:MB_gco} 
     
    647602%} 
    648603 
    649 % ------------------------------------------------------------------------------------------------------------- 
    650 % The s-coordinate Formulation 
    651 % ------------------------------------------------------------------------------------------------------------- 
     604%% ================================================================================================= 
    652605\subsection{\textit{S}-coordinate formulation} 
    653606 
     
    737690} 
    738691 
    739 % ------------------------------------------------------------------------------------------------------------- 
    740 % Curvilinear \zstar-coordinate System 
    741 % ------------------------------------------------------------------------------------------------------------- 
     692%% ================================================================================================= 
    742693\subsection{Curvilinear \zstar-coordinate system} 
    743694\label{subsec:MB_zco_star} 
    744695 
    745 %>>>>>>>>>>>>>>>>>>>>>>>>>>>> 
    746696\begin{figure}[!b] 
    747697  \centering 
     
    754704  \label{fig:MB_z_zstar} 
    755705\end{figure} 
    756 %>>>>>>>>>>>>>>>>>>>>>>>>>>>> 
    757706 
    758707In this case, the free surface equation is nonlinear, and the variations of volume are fully taken into account. 
     
    827776%end MOM doc %%% 
    828777 
    829 \newpage 
    830  
    831 % ------------------------------------------------------------------------------------------------------------- 
    832 % Terrain following  coordinate System 
    833 % ------------------------------------------------------------------------------------------------------------- 
     778%% ================================================================================================= 
    834779\subsection{Curvilinear terrain-following \textit{s}--coordinate} 
    835780\label{subsec:MB_sco} 
    836781 
    837 % ------------------------------------------------------------------------------------------------------------- 
    838 % Introduction 
    839 % ------------------------------------------------------------------------------------------------------------- 
     782%% ================================================================================================= 
    840783\subsubsection{Introduction} 
    841784 
     
    918861It also offers a completely general transformation, $s=s(i,j,z)$ for the vertical coordinate. 
    919862 
    920 % ------------------------------------------------------------------------------------------------------------- 
    921 % Curvilinear z-tilde coordinate System 
    922 % ------------------------------------------------------------------------------------------------------------- 
     863%% ================================================================================================= 
    923864\subsection{\texorpdfstring{Curvilinear \ztilde-coordinate}{}} 
    924865\label{subsec:MB_zco_tilde} 
     
    929870Its use is therefore not recommended. 
    930871 
    931 \newpage 
    932  
    933 % ================================================================ 
    934 % Subgrid Scale Physics 
    935 % ================================================================ 
     872%% ================================================================================================= 
    936873\section{Subgrid scale physics} 
    937874\label{sec:MB_zdf_ldf} 
     
    957894The formulation of these terms and their underlying physics are briefly discussed in the next two subsections. 
    958895 
    959 % ------------------------------------------------------------------------------------------------------------- 
    960 % Vertical Subgrid Scale Physics 
    961 % ------------------------------------------------------------------------------------------------------------- 
     896%% ================================================================================================= 
    962897\subsection{Vertical subgrid scale physics} 
    963898\label{subsec:MB_zdf} 
     
    991926The choices available in \NEMO\ are discussed in \autoref{chap:ZDF}). 
    992927 
    993 % ------------------------------------------------------------------------------------------------------------- 
    994 % Lateral Diffusive and Viscous Operators Formulation 
    995 % ------------------------------------------------------------------------------------------------------------- 
     928%% ================================================================================================= 
    996929\subsection{Formulation of the lateral diffusive and viscous operators} 
    997930\label{subsec:MB_ldf} 
     
    1047980and UBS advection schemes when flux form is chosen for the momentum advection. 
    1048981 
     982%% ================================================================================================= 
    1049983\subsubsection{Lateral laplacian tracer diffusive operator} 
    1050984 
     
    10881022while in $s$-coordinates $\pd[]{k}$ is replaced by $\pd[]{s}$. 
    10891023 
     1024%% ================================================================================================= 
    10901025\subsubsection{Eddy induced velocity} 
    10911026 
     
    11241059The latter strategy is used in \NEMO\ (cf. \autoref{chap:LDF}). 
    11251060 
     1061%% ================================================================================================= 
    11261062\subsubsection{Lateral bilaplacian tracer diffusive operator} 
    11271063 
     
    11351071the harmonic eddy diffusion coefficient set to the square root of the biharmonic one. 
    11361072 
     1073%% ================================================================================================= 
    11371074\subsubsection{Lateral Laplacian momentum diffusive operator} 
    11381075 
     
    11671104a geographical coordinate system \citep{lengaigne.madec.ea_JGR03}. 
    11681105 
     1106%% ================================================================================================= 
    11691107\subsubsection{Lateral bilaplacian momentum diffusive operator} 
    11701108 
  • NEMO/trunk/doc/latex/NEMO/subfiles/chap_model_basics_zstar.tex

    r11584 r11596  
    22 
    33\begin{document} 
    4 % ================================================================ 
    5 % Chapter 1 Model Basics 
    6 % ================================================================ 
    7 % ================================================================ 
    8 % Curvilinear \zstar- \sstar-coordinate System 
    9 % ================================================================ 
    104\chapter{ essai \zstar \sstar} 
    115\section{Curvilinear \zstar- or \sstar coordinate system} 
    12  
    13 % ------------------------------------------------------------------------------------------------------------- 
    14 % ???? 
    15 % ------------------------------------------------------------------------------------------------------------- 
    166 
    177\colorbox{yellow}{ to be updated } 
     
    7060%%% 
    7161 
    72 % ================================================================ 
    73 % Surface Pressure Gradient and Sea Surface Height 
    74 % ================================================================ 
    7562\section[Surface pressure gradient and sea surface heigth (\textit{dynspg.F90})]{Surface pressure gradient and sea surface heigth (\protect\mdl{dynspg})} 
    7663\label{sec:MBZ_dyn_hpg_spg} 
  • NEMO/trunk/doc/latex/NEMO/subfiles/chap_time_domain.tex

    r11584 r11596  
    33\begin{document} 
    44 
    5 % ================================================================ 
    6 % Chapter 2 ——— Time Domain (step.F90) 
    7 % ================================================================ 
    85\chapter{Time Domain} 
    96\label{chap:TD} 
     
    1613  would help  ==> to be added} 
    1714%%%% 
    18  
    19 \newpage 
    2015 
    2116Having defined the continuous equations in \autoref{chap:MB}, we need now to choose a time discretization, 
     
    2520the consequences for the order in which the equations are solved. 
    2621 
    27 % ================================================================ 
    28 % Time Discretisation 
    29 % ================================================================ 
    3022\section{Time stepping environment} 
    3123\label{sec:TD_environment} 
     
    5547The 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. 
    5648 
    57 % ------------------------------------------------------------------------------------------------------------- 
    58 %        Non-Diffusive Part---Leapfrog Scheme 
    59 % ------------------------------------------------------------------------------------------------------------- 
    6049\section{Non-diffusive part --- Leapfrog scheme} 
    6150\label{sec:TD_leap_frog} 
     
    9887filter parameter and the viscosity and diffusion coefficients. 
    9988 
    100 % ------------------------------------------------------------------------------------------------------------- 
    101 %        Diffusive Part---Forward or Backward Scheme 
    102 % ------------------------------------------------------------------------------------------------------------- 
    10389\section{Diffusive part --- Forward or backward scheme} 
    10490\label{sec:TD_forward_imp} 
     
    165151(see for example \citet{richtmyer.morton_bk67}). 
    166152 
    167 % ------------------------------------------------------------------------------------------------------------- 
    168 %        Surface Pressure gradient 
    169 % ------------------------------------------------------------------------------------------------------------- 
    170153\section{Surface pressure gradient} 
    171154\label{sec:TD_spg_ts} 
     
    184167 
    185168%\gmcomment{ 
    186 %>>>>>>>>>>>>>>>>>>>>>>>>>>>> 
    187169\begin{figure}[!t] 
    188170  \centering 
     
    197179  \label{fig:TD_TimeStep_flowchart} 
    198180\end{figure} 
    199 %>>>>>>>>>>>>>>>>>>>>>>>>>>>> 
    200181%} 
    201182 
    202 % ------------------------------------------------------------------------------------------------------------- 
    203 %        The Modified Leapfrog -- Asselin Filter scheme 
    204 % ------------------------------------------------------------------------------------------------------------- 
    205183\section{Modified Leapfrog -- Asselin filter scheme} 
    206184\label{sec:TD_mLF} 
     
    245223even if separated by only $\rdt$ since the time filter is no longer applied to the forcing term. 
    246224 
    247 %>>>>>>>>>>>>>>>>>>>>>>>>>>>> 
    248225\begin{figure}[!t] 
    249226  \centering 
     
    261238  \label{fig:TD_MLF_forcing} 
    262239\end{figure} 
    263 %>>>>>>>>>>>>>>>>>>>>>>>>>>>> 
    264  
    265 % ------------------------------------------------------------------------------------------------------------- 
    266 %        Start/Restart strategy 
    267 % ------------------------------------------------------------------------------------------------------------- 
     240 
    268241\section{Start/Restart strategy} 
    269242\label{sec:TD_rst} 
     
    315288%------------------------------------------------------------------------------------------------------------- 
    316289%        Time Domain 
    317 % ------------------------------------------------------------------------------------------------------------- 
    318 \subsection{Time domain} 
    319 \label{subsec:TD_time} 
    320 %--------------------------------------------namrun------------------------------------------- 
    321  
    322 %-------------------------------------------------------------------------------------------------------------- 
    323  
    324 Options are defined through the  \nam{dom}{dom} namelist variables. 
    325  \colorbox{yellow}{add here a few word on nit000 and nitend} 
    326  
    327  \colorbox{yellow}{Write documentation on the calendar and the key variable adatrj} 
    328  
    329 add a description of daymod, and the model calandar (leap-year and co) 
    330  
    331 }        %% end add 
    332  
    333  
    334  
    335 %% 
    336 \gmcomment{       % add implicit in vvl case  and Crant-Nicholson scheme 
    337  
    338 Implicit time stepping in case of variable volume thickness. 
    339  
    340 Tracer case (NB for momentum in vector invariant form take care!) 
    341  
    342 \begin{flalign*} 
    343   &\frac{\lt( e_{3t}\,T \rt)_k^{t+1}-\lt( e_{3t}\,T \rt)_k^{t-1}}{2\rdt} 
    344   \equiv \text{RHS}+ \delta_k \lt[ {\frac{A_w^{vt} }{e_{3w}^{t+1} }\delta_{k + 1/2} \lt[ {T^{t+1}} \rt]} 
    345   \rt]      \\ 
    346   &\lt( e_{3t}\,T \rt)_k^{t+1}-\lt( e_{3t}\,T \rt)_k^{t-1} 
    347   \equiv {2\rdt} \ \text{RHS}+ {2\rdt} \ \delta_k \lt[ {\frac{A_w^{vt} }{e_{3w}^{t+1} }\delta_{k + 1/2} \lt[ {T^{t+1}} \rt]} 
    348   \rt]      \\ 
    349   &\lt( e_{3t}\,T \rt)_k^{t+1}-\lt( e_{3t}\,T \rt)_k^{t-1} 
    350   \equiv 2\rdt \ \text{RHS} 
    351   + 2\rdt \ \lt\{ \lt[ \frac{A_w^{vt}}{e_{3w}^{t+1}} \rt]_{k + 1/2} [ T_{k +1}^{t+1} - T_k      ^{t+1} ] 
    352     - \lt[ \frac{A_w^{vt}}{e_{3w}^{t+1}} \rt]_{k - 1/2} [ T_k       ^{t+1} - T_{k -1}^{t+1} ]  \rt\}     \\ 
    353   &\\ 
    354   &\lt( e_{3t}\,T \rt)_k^{t+1} 
    355   -  {2\rdt} \           \lt[ \frac{A_w^{vt}}{e_{3w}^{t+1}} \rt]_{k + 1/2}                  T_{k +1}^{t+1} 
    356   + {2\rdt} \ \lt\{  \lt[ \frac{A_w^{vt}}{e_{3w}^{t+1}} \rt]_{k + 1/2} 
    357     +  \lt[ \frac{A_w^{vt}}{e_{3w}^{t+1}} \rt]_{k - 1/2}     \rt\}   T_{k    }^{t+1} 
    358   -  {2\rdt} \           \lt[ \frac{A_w^{vt}}{e_{3w}^{t+1}} \rt]_{k - 1/2}                  T_{k -1}^{t+1}      \\ 
    359   &\equiv \lt( e_{3t}\,T \rt)_k^{t-1} + {2\rdt} \ \text{RHS}    \\ 
    360   % 
    361 \end{flalign*} 
    362 \begin{flalign*} 
    363   \allowdisplaybreaks 
    364   \intertext{ Tracer case } 
    365   % 
    366   &  \qquad \qquad  \quad   -  {2\rdt}                  \ \lt[ \frac{A_w^{vt}}{e_{3w}^{t+1}} \rt]_{k + 1/2} 
    367   \qquad \qquad \qquad  \qquad  T_{k +1}^{t+1}   \\ 
    368   &+ {2\rdt} \ \biggl\{  (e_{3t})_{k   }^{t+1}  \bigg. +    \lt[ \frac{A_w^{vt}}{e_{3w}^{t+1}} \rt]_{k + 1/2} 
    369   +   \lt[ \frac{A_w^{vt}}{e_{3w}^{t+1}} \rt]_{k - 1/2} \bigg. \biggr\}  \ \ \ T_{k   }^{t+1}  &&\\ 
    370   & \qquad \qquad  \qquad \qquad \qquad \quad \ \ -  {2\rdt} \                          \lt[ \frac{A_w^{vt}}{e_{3w}^{t+1}} \rt]_{k - 1/2}                          \quad \ \ T_{k -1}^{t+1} 
    371   \ \equiv \ \lt( e_{3t}\,T \rt)_k^{t-1} + {2\rdt} \ \text{RHS}  \\ 
    372   % 
    373 \end{flalign*} 
    374 \begin{flalign*} 
    375   \allowdisplaybreaks 
    376   \intertext{ Tracer content case } 
    377   % 
    378   & -  {2\rdt} \              & \frac{(A_w^{vt})_{k + 1/2}} {(e_{3w})_{k + 1/2}^{t+1}\;(e_{3t})_{k +1}^{t+1}}  && \  \lt( e_{3t}\,T \rt)_{k +1}^{t+1}   &\\ 
    379   & + {2\rdt} \ \lt[ 1  \rt.+ & \frac{(A_w^{vt})_{k + 1/2}} {(e_{3w})_{k + 1/2}^{t+1}\;(e_{3t})_k^{t+1}} 
    380   + & \frac{(A_w^{vt})_{k - 1/2}} {(e_{3w})_{k - 1/2}^{t+1}\;(e_{3t})_k^{t+1}}  \lt.  \rt]  & \lt( e_{3t}\,T \rt)_{k   }^{t+1}  &\\ 
    381   & -  {2\rdt} \               & \frac{(A_w^{vt})_{k - 1/2}} {(e_{3w})_{k - 1/2}^{t+1}\;(e_{3t})_{k -1}^{t+1}}     &\  \lt( e_{3t}\,T \rt)_{k -1}^{t+1} 
    382   \equiv \lt( e_{3t}\,T \rt)_k^{t-1} + {2\rdt} \ \text{RHS}  & 
    383 \end{flalign*} 
    384  
    385 %% 
    386 } 
    387  
    388 \onlyinsubfile{\input{../../global/epilogue}} 
    389  
    390 \end{document} 
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