Index: /branches/UKMO/dev_r5518_GO6_under_ice_relax_dr_hook/ADM/DOC_SCRIPTS/extract_rst.sh
===================================================================
--- /branches/UKMO/dev_r5518_GO6_under_ice_relax_dr_hook/ADM/DOC_SCRIPTS/extract_rst.sh (revision 11643)
+++ /branches/UKMO/dev_r5518_GO6_under_ice_relax_dr_hook/ADM/DOC_SCRIPTS/extract_rst.sh (revision 11643)
@@ -0,0 +1,312 @@
+#! /bin/sh
+set -x
+#+
+#
+# ==========
+# extract.sh
+# ==========
+#
+# -----------------------------------------
+# extract ReStructuredText from source file
+# -----------------------------------------
+#
+# SYNOPSIS
+# ========
+#
+# ``extract_rst.sh -i filein -l language -o fileout``
+#
+# DESCRIPTION
+# ===========
+#
+# ``extract_rst.sh`` extracts ReST comments from the file given in argument
+#
+# -i input file
+# -o output file (ReST)
+# -l language
+#
+# Comment block (start, end) identification depends on language :
+#
+# *F90*
+# FORTRAN source free form
+# *fortran*
+# FORTRAN source fixed form
+# *sh*
+# shell scripts
+# *IDL*
+# IDL source
+# *xml*
+# XML and XSL
+# *dot*
+# graphviz files
+# *php*
+# PHP files
+# *matlab*
+# matlab or octave files
+#
+# EXAMPLES
+# ========
+#
+# To extract ReST comments of this shell script::
+#
+# $ extract_rst.sh -i extract_rst.sh -l sh -o extract_rst.sh.rst
+# iii : rst lines of extract_rst.sh are in extract_rst.sh.rst
+#
+# You can produce HTML file from this new file::
+#
+# $ rst2html.py --input-encoding=ISO-8859-15 extract_rst.sh.rst \
+# /usr/temp/${LOGNAME}/extract_rst.sh.html
+#
+# You can produce PDF file from this new file::
+#
+# $ rst2newlatex.py --input-encoding=ISO-8859-15 extract_rst.sh.rst \
+# /usr/temp/${LOGNAME}/extract_rst.sh.tex
+# $ pdflatex extract_rst.sh.tex
+#
+# Of course beware of consistency of path on links.
+#
+# CAUTIONS
+# ========
+#
+# Becaue of poor implementation of Standard FORTRAN in cpp (prepocessing)
+# within gfortran and g95, ReST comments might induce trouble in
+# FORTRAN sources.
+#
+# For example following line is pointed out be gfortran with
+# ``error: unterminated comment``.
+# This is because ``/*`` is the beginning of a C style comment !!
+# ::
+#
+# ! **MEAN** = sum( *X*\ (:) )/*ntime*
+#
+#
+#
+# One can modify this ReST line with
+# ::
+#
+# ! **MEAN** = sum( *X*\ (:) ) / *ntime*
+#
+# TODO
+# ====
+#
+# check parameters
+#
+# log
+#
+# add perl
+#
+# SEE ALSO
+# ========
+#
+# ReStructuredText_
+#
+# .. _ReStructuredText: http://docutils.sourceforge.net/rst.html
+#
+# Docutils_
+#
+# .. _Docutils: http://docutils.sourceforge.net/
+#
+# EVOLUTIONS
+# ==========
+#
+# $Id: extract_rst.sh 2246 2010-10-13 09:47:23Z rblod $
+#
+# - fplod 2009-04-20T08:13:37Z aedon.locean-ipsl.upmc.fr (Darwin)
+#
+# * add CAUTIONS paragraph to warn about possible FORTRAN compiling problem
+#
+# - fplod 2009-04-03T14:53:18Z aedon.locean-ipsl.upmc.fr (Darwin)
+#
+# * usage of tr instead of sed to remove ``\r``
+# due to difference between ``/sw/bin/sed`` and ``/usr/bin/sed`` (the last
+# one do not work coorectly on ``\r`` interpertation ie: remove the first occurence of
+# ``r``)
+#
+# - fplod 2009-02-10T10:46:23Z aedon.locean-ipsl.upmc.fr (Darwin)
+#
+# * add language fortran for FORTRAN source in fixed form
+#
+# - fplod 2009-01-05T11:41:33Z aedon.locean-ipsl.upmc.fr (Darwin)
+#
+# * remove \\r (CRLF) from file before awk and sed (otherwise ReST block
+# was not found in "ISO-8859 text, with CRLF line terminators" files
+#
+# - fplod 2008-12-22T10:37:37Z aedon.locean-ipsl.upmc.fr (Darwin)
+#
+# * add matlab (octave)
+#
+# - fplod 2008-09-17T13:40:37Z aedon.locean-ipsl.upmc.fr (Darwin)
+#
+# * add php language
+#
+# - fplod 2008-09-08T09:34:04Z aedon.locean-ipsl.upmc.fr (Darwin)
+#
+# * add F90 language
+#
+# - fplod 2008-08-08T08:28:30Z aedon.locean-ipsl.upmc.fr (Darwin)
+#
+# * add KML files (with other XML files)
+# * add parameters ``-i`` ``-l`` ``-o``
+#
+# - fplod 200807
+#
+# * creation
+#
+#-
+#
+system=$(uname)
+case "${system}" in
+ AIX|IRIX64)
+ echo " www : no specific posix checking"
+ ;;
+ *)
+ set -o posix
+ ;;
+esac
+unset system
+#
+command=$(basename ${0})
+log_date=$(date -u +"%Y%m%dT%H%M%SZ")
+log=/tmp/$(basename ${command} .sh).log.${log_date}
+#
+usage=" Usage : ${command} -i filein -l language -o fileout"
+#
+minargcount=6
+#echo " narg ${#}"
+if [ ${#} -lt ${minargcount} ]
+then
+ echo "eee : not enought arguments"
+ echo "${usage}"
+ exit 1
+fi
+#
+# default
+# n.a.
+#
+while [ ! -z "${1}" ]
+do
+ case ${1} in
+ -i)
+ filein=${2}
+ shift
+ ;;
+ -o)
+ fileout=${2}
+ shift
+ ;;
+ -l)
+ language=${2}
+ shift
+ ;;
+ -h)
+ echo "${usage}"
+ exit 0
+ ;;
+ *)
+ echo "eee : unknown option ${1}"
+ echo "${usage}"
+ exit 1
+ ;;
+ esac
+ # next flag
+ shift
+done
+#
+set -u
+#
+# ++ check param
+#
+case "${language}" in
+ fortran)
+ awkblockstart="^C\+$"
+ awkblockend="^C-$"
+ sedblockstart="^C+$"
+ sedblockend="^C-$"
+ comment="^C"
+ ;;
+ F90)
+ awkblockstart="^!\+$"
+ awkblockend="^!-$"
+ sedblockstart="^!+$"
+ sedblockend="^!-$"
+ comment="^!"
+ ;;
+ IDL)
+ awkblockstart="^;\+$"
+ awkblockend="^;-$"
+ sedblockstart="^;+$"
+ sedblockend="^;-$"
+ comment="^;"
+ ;;
+ xml)
+ awkblockstart="^$"
+ sedblockstart="^$"
+ comment=""
+ ;;
+ sh)
+ # iii : awk '/^\#\+/,/^\#\-/' $file
+ awkblockstart="^\#\+$"
+ awkblockend="^\#\-$"
+ sedblockstart="^#+"
+ sedblockend="^#-"
+ comment="^#"
+ ;;
+ dot|php)
+ awkblockstart="^\/\*rst$"
+ awkblockend="*\/"
+ sedblockstart="^\/\*rst$"
+ sedblockend="^\*\/"
+ comment=""
+ ;;
+ matlab)
+ awkblockstart="^%\+$"
+ awkblockend="^%-$"
+ sedblockstart="^%+$"
+ sedblockend="^%-$"
+ comment="^%"
+ ;;
+ *)
+ echo "eee : ${language} not implemented"
+ exit 1
+ ;;
+esac
+#
+# just in case suppress \r at the end of lines
+tr -d '\r' < ${filein} > /tmp/${$}_0
+#
+# put rst blocks in one temporary file
+#awk '/^;+/,/^;-/' a.pro | sed -e "/^;+$/d" -e "/^;-$/d" -e "s/^;//"
+cmdawk="awk '/${awkblockstart}/,/${awkblockend}/' /tmp/${$}_0 > /tmp/${$}_1" #++
+eval ${cmdawk}
+if [ ! -s /tmp/${$}_1 ]
+then
+ rm /tmp/${$}_0 /tmp/${$}_1
+ echo "iii : no rst comments in ${filein}"
+ exit 1
+fi
+#
+# suppress begin and end of each block
+sedcmd="sed -e \"/${sedblockstart}/d\" -e \"/${sedblockend}/d\" /tmp/${$}_1 > /tmp/${$}_2"
+eval ${sedcmd}
+#
+# suppress comment at the beginning of each line
+if [ "${comment}" != "" ]
+then
+ sedcmd="sed -e \"s/${comment}//\" /tmp/${$}_2 > /tmp/${$}_3"
+ eval ${sedcmd}
+ # suppress first blank
+ cp /tmp/${$}_3 /tmp/${$}_2
+ sed -e "s/^ //" /tmp/${$}_2 > /tmp/${$}_3
+ cp /tmp/${$}_3 ${fileout}
+else
+ cp /tmp/${$}_2 ${fileout}
+fi
+#
+echo "iii : rst lines of ${filein} are in ${fileout}"
+#
+# clean
+rm /tmp/${$}_0 /tmp/${$}_1 /tmp/${$}_2 /tmp/${$}_3 2> /dev/null
+#
+# exit
+exit 0
Index: /branches/UKMO/dev_r5518_GO6_under_ice_relax_dr_hook/ADM/DOC_SCRIPTS/install.sh
===================================================================
--- /branches/UKMO/dev_r5518_GO6_under_ice_relax_dr_hook/ADM/DOC_SCRIPTS/install.sh (revision 11643)
+++ /branches/UKMO/dev_r5518_GO6_under_ice_relax_dr_hook/ADM/DOC_SCRIPTS/install.sh (revision 11643)
@@ -0,0 +1,193 @@
+#!/bin/sh -x
+#+
+#
+# ==========
+# install.sh
+# ==========
+#
+# ----------------------------------------------
+# publication of HTML files and associated files
+# ----------------------------------------------
+#
+# SYNOPSIS
+# ========
+#
+# ::
+#
+# $ install.sh -w dirwww -p dirpublish -u urlpublish -l login
+#
+# DESCRIPTION
+# ===========
+#
+# publication (rsync) of dirwww content on dirpublish given in argument
+#
+# If the host of publication is cerbere.locean-ipsl.upmc.fr, a specific update
+# is launched.
+#
+# -w input directort
+# -p output directory
+# -u output url
+# -l login used to access on output url
+#
+# If needed, existing directories might be removed before (to erase obsolete
+# files), using ncftp tool :
+#
+# For example, to clean directory on LOCEAN web server ::
+#
+# $ ncftp -u fplod www.locean-ipsl.upmc.fr
+# ncftp> cd fplod
+# ncftp> rm -f pageperso/.DS_Store
+# ncftp> rm -rf pageperso
+# ncftp> exit
+#
+# EXAMPLES
+# ========
+#
+# EVOLUTIONS
+# ==========
+#
+# $Id: install.sh 2246 2010-10-13 09:47:23Z rblod $
+#
+# - fplod 2008-09-16T15:24:26Z aedon.locean-ipsl.upmc.fr (Darwin)
+#
+# * comments in ReStructured Text
+#
+# - fplod 2008-06-17T09:10:19Z aedon.locean-ipsl.upmc.fr (Darwin)
+#
+# * add -l parameter only used in specific case at LOCEAN when user
+# parameter of persoweb must be different tthan login (ex: acmo vs fplod)
+# * replace http://www.lodyc.jussieu.fr/info_reseau/persoweb/?fastupdate=1&user=${user}" by
+# http://intranet.locean-ipsl.upmc.fr/persoweb/?fastupdate=1&user=${user}
+#
+# - fplod 2008-03-28T10:26:58Z aedon.locean-ipsl.upmc.fr (Darwin)
+#
+# * new personnal webpages policy at LOCEAN so new command and new parameter (-u)
+#
+# - fplod 2007-09-28T09:30:43Z aedon.locean-ipsl.upmc.fr (Darwin)
+#
+# * parametrisation and translation
+#
+# - smasson 2007-06-07T16:43:42Z arete.locean-ipsl.upmc.fr (Darwin)
+#
+# * can give the answer with input parameters
+#
+# - fplod 2007-04-26T11:51:42Z aedon.locean-ipsl.upmc.fr (Darwin)
+#
+#-
+system=$(uname)
+case "${system}" in
+ AIX|IRIX64)
+ echo " www : no specific posix checking"
+ ;;
+ *)
+ set -o posix
+ ;;
+esac
+unset system
+#
+command=$(basename ${0})
+log_date=$(date -u +"%Y%m%dT%H%M%SZ")
+log=/tmp/$(basename ${command} .sh).log.${log_date}
+#
+usage=" Usage : ${command} -w dirwww -p dirpublish -u urlpublish -l login"
+#
+minargcount=4
+#echo " narg ${#}"
+if [ ${#} -lt ${minargcount} ]
+then
+ echo "eee : not enought arguments"
+ echo "${usage}"
+ exit 1
+fi
+unset minargcount
+#
+# default
+dirpublish="none"
+urlpublish="none"
+login="none"
+#
+while [ ! -z "${1}" ]
+do
+ case ${1} in
+ -w)
+ dirwww=${2}
+ shift
+ ;;
+ -p)
+ dirpublish=${2}
+ shift
+ ;;
+ -u)
+ urlpublish=${2}
+ shift
+ ;;
+ -l)
+ login=${2}
+ shift
+ ;;
+ esac
+ # next flag
+ shift
+done
+#
+set -u
+#
+# ++ check directories
+#
+answer=${1:-" "}
+case ${answer} in
+ y|Y|n|N)
+ ;;
+ *)
+ if [ "${dirpublish}" != "none" ]
+ then
+ echo "Do you want to install on ${dirpublish} (y|[n]) ?"
+ read answer
+ fi
+ if [ "${urlpublish}" != "none" ]
+ then
+ echo "Do you want to install on ${urlpublish} (y|[n]) ?"
+ read answer
+ fi
+ ;;
+esac
+#
+case ${answer} in
+ y|Y)
+ if [ "${dirpublish}" != "none" ]
+ then
+ # copy of ${dirwww} on $dirpublish
+ echo "iii : update of ${dirpublish}"
+ rsync -av --exclude=".DS_Store" -e ssh ${dirwww}/ ${dirpublish}
+ # detect if in dirpublish following this pattern [USER@]HOST:SRC, HOST
+ # is cerbere.locean-ipsl.upmc.fr. If so, a specific update is launched
+ userhost=${dirpublish%%:*}
+ host=${userhost##*@}
+ if [ ${login} = "none" ]
+ then
+ user=${userhost%%@*}
+ else
+ user=${login}
+ fi
+ if [ "${host}" = "cerbere.locean-ipsl.upmc.fr" ]
+ then
+ wget -q "http://intranet.locean-ipsl.upmc.fr/persoweb/?fastupdate=1&user=${user}" -O /dev/null
+ fi
+ else
+ # urlpublish=http://www.locean-ipsl.upmc.fr/~ginette/produit
+ dirpublish=${urlpublish##*~}
+ cd ${dirwww}
+ #lftp -e "mirror -R . ${dirpublish};quit" -u ${LOGNAME} skyros.locean-ipsl.upmc.fr
+ lftp -e "mirror -R . ${dirpublish};quit" -u ${LOGNAME} localhost
+ # pour acmo a la main ++
+ #++lftp -e 'mirror -R . acmo/nouveaux/;quit' -u fplod www.locean-ipsl.upmc.fr
+ # ++ log
+ fi
+ ;;
+ *)
+ echo "no update of ${dirpublish} or ${urlpublish}"
+ ;;
+esac
+#
+# normal exit
+exit 0
Index: /branches/UKMO/dev_r5518_GO6_under_ice_relax_dr_hook/ADM/DOC_SCRIPTS/makefile_compile
===================================================================
--- /branches/UKMO/dev_r5518_GO6_under_ice_relax_dr_hook/ADM/DOC_SCRIPTS/makefile_compile (revision 11643)
+++ /branches/UKMO/dev_r5518_GO6_under_ice_relax_dr_hook/ADM/DOC_SCRIPTS/makefile_compile (revision 11643)
@@ -0,0 +1,335 @@
+#+
+#
+# ========
+# makefile
+# ========
+#
+# -----------------------------------------------
+# generation of documentation of NEMO compilation
+# -----------------------------------------------
+#
+# TODO
+# ====
+#
+# add -W to sphinx command when encoding problems are solved
+#
+# usage of sphinx/source/Makefile
+#
+# revision of manual section of php and xsl
+#
+# EVOLUTIONS
+# ==========
+#
+# $Id: makefile_compile 2520 2010-12-27 14:43:36Z rblod $
+#
+# - fplod 20100419T145702Z aedon.locean-ipsl.upmc.fr (Darwin)
+#
+# * remove rest2web (sphinx is prefered)
+#
+# - fplod 20100323T135104Z aedon.locean-ipsl.upmc.fr (Darwin)
+#
+# * remove one pdf and html
+#
+# - fplod 20100311T143131Z aedon.locean-ipsl.upmc.fr (Darwin)
+#
+# * add rest2web (alternative to sphinx)
+#
+# - fplod 20100310T190253Z aedon.locean-ipsl.upmc.fr (Darwin)
+#
+# * add php and xsl files to man_troff
+#
+# - fplod 20100310T182201Z aedon.locean-ipsl.upmc.fr (Darwin)
+#
+# * usage of sphinx (for the first time !) not yet ok ...
+#
+# - fplod 20100310T091541Z aedon.locean-ipsl.upmc.fr (Darwin)
+#
+# * add man_troff with shell scripts possible now with docutils 0.6
+# can be test with
+# $ man -M ../doc//manuals/man bibopa.sh
+#
+# - fplod 2009-05-13T14:08:49Z aedon.locean-ipsl.upmc.fr (Darwin)
+#
+# * implicit rules
+#
+# nb : may be will only work wih GNU make
+# but easier to update : only one line to add in thi makefile when
+# a new file is added in $(DIRSRC) directory
+#
+# * rst2latex usage of manuals_many.sty and manual_one.sty (for TOC and parindent)
+# * bug fix for PDF manual (missing one pdf2latex)
+#
+# - fplod 2008-10-28T10:59:44Z aedon.locean-ipsl.upmc.fr (Darwin)
+#
+# * add newpage directive
+# (thanks to http://docutils.sourceforge.net/docs/user/latex.html)
+#
+# - fplod 2008-09-17T09:16:08Z aedon.locean-ipsl.upmc.fr (Darwin)
+# * add xsl files
+#
+# - fplod 2008-09-16T14:59:02Z aedon.locean-ipsl.upmc.fr (Darwin)
+# * creation
+#
+# SEE ALSO
+# ========
+#
+# extract_rst.sh_
+#
+# .. _extract_rst.sh: ../extract_rst.sh.html
+#
+#-
+#
+PRODUCT = \
+NEMO_UTIL
+
+PRODUCTNAME = \
+$$(echo $(PRODUCT) | tr [:lower:] [:upper:])
+
+DIRSRC = \
+../../NEMOGCM/TOOLS/COMPILE
+
+DIRADM = \
+./
+
+DIRTMP = \
+./
+
+DIRDESIGN = \
+./design/
+
+DIRWWW = \
+./doc/
+
+URLPUBLISH = \
+http://192.168.0.12/~rblod/$(PRODUCT)
+
+LIST_SRCSH = $(wildcard $(DIRSRC)/*.sh) \
+$(DIRSRC)/../../CONFIG/makenemo \
+$(DIRSRC)/../maketools
+
+LIST_SRCSH_RST = $(addprefix $(DIRTMP)/, $(notdir $(addsuffix .rst,$(LIST_SRCSH))))
+
+LIST_SRCSH_R2W = $(addprefix $(DIRTMP)/rest2web_tmpdir/, $(notdir $(addsuffix .txt,$(LIST_SRCSH))))
+
+LIST_SRCSH_TROFF = $(addprefix $(DIRWWW)/manuals/man/man1/, $(notdir $(addsuffix .1,$(LIST_SRCSH))))
+
+LIST_SRCSH_HTML = $(adprefix $(DIRWWW)/manuals/html/many/, $(notdir $(addsuffix .html,$(LIST_SRCSH))))
+
+LIST_SRCSH_PDF = $(addprefix $(DIRSRC)+$(DIRWWW)/manuals/pdf/many/,$(notdir $(addsuffix .pdf,$(LIST_SRCSH))))
+
+RST2MAN = \
+rst2man.py
+
+RST2HTML = \
+rst2html.py
+
+RST2LATEX = \
+rst2latex.py
+
+RST2LATEX_OPTIONS_MANY = \
+--documentclass=article \
+--stylesheet=manuals_many.sty \
+--traceback \
+--use-verbatim-when-possible
+
+.PHONY : \
+help \
+before \
+clean \
+cleantmp \
+design \
+htmllinkcheckb \
+htmllinkchecka \
+spellcheck \
+all \
+man \
+man_troff \
+man_html \
+man_html_many \
+man_html_sphinx \
+man_pdf \
+man_pdf_many \
+man_pdf_sphinx
+
+help :
+ @echo "Prepare output directories :"
+ @echo "\$$ make before"
+ @echo ""
+ @echo "Following commands are available to build outputs :"
+ @echo "\$$ make all"
+ @echo " "
+ @echo "Check links before installation : "
+ @echo "\$$ make htmllinkcheckb"
+ @echo " "
+ @echo "Last step = installation"
+ @echo "\$$ make install"
+ @echo " "
+ @echo "Check links after installation : "
+ @echo "\$$ make htmllinkchecka"
+ @echo " "
+
+before :
+ @mkdir -p $(DIRWWW)/manuals/man/man1/
+ @mkdir -p $(DIRWWW)/manuals/html/many/
+ @mkdir -p $(DIRWWW)/manuals/html/rest2web/
+ @mkdir -p $(DIRWWW)/manuals/html/rest2web/css/
+ @mkdir -p $(DIRWWW)/manuals/html/sphinx/
+ @mkdir -p $(DIRWWW)/manuals/pdf/many/
+ @mkdir -p $(DIRWWW)/manuals/pdf/sphinx/
+ @mkdir -p $(DIRTMP)/sphinx_tmpdir/doctrees/
+ @mkdir -p $(DIRTMP)/rest2web_tmpdir/
+
+install :
+ @install.sh -w $(DIRWWW) -u $(URLPUBLISH)
+
+clean : \
+cleantmp
+ -@rm -fr $(DIRWWW)/
+ -@rm -fr $(DIRWWW)/manuals/man/
+ -@rm -fr $(DIRWWW)/manuals/html/
+ -@rm -fr $(DIRWWW)/manuals/pdf/
+
+cleantmp :
+ -@rm -f $(DIRTMP)/all.xml
+ -@rm -f $(DIRTMP)/*.rst
+ -@rm -f $(DIRTMP)/*.tex
+ -@rm -rf $(DIRTMP)/rest2web_tmpdir/
+ -@rm -f $(DIRTMP)/rest2web.log
+ -@rm -f $(DIRWWW)/manuals/pdf/many/*.aux
+ -@rm -f $(DIRWWW)/manuals/pdf/many/*.log
+ -@rm -f $(DIRWWW)/manuals/pdf/many/*.out
+ -@rm -rf $(DIRTMP)/sphinx_tmpdir/
+ -@rm -f $(DIRTMP)/sphinx_*.log
+
+design : \
+$(DIRDESIGN)/images/$(PRODUCT)_fulldependencies.png \
+$(DIRDESIGN)/images/$(PRODUCT)_fulldependencies.svg
+
+htmllinkcheckb :
+ @linkchecker.sh -d $(DIRWWW)/manuals/html/
+
+htmllinkchecka :
+ @linkchecker.sh -u $(URLPUBLISH)
+
+spellcheck :
+ @++aspell --mode=sgml --master=francais -c \
+ $(DIRSRC)/$(PRODUCT).xml
+
+all : \
+SPECIAL_RST \
+man_troff \
+man_html \
+man_pdf
+
+man_troff : \
+$(LIST_SRCSH_TROFF)
+
+man_html : \
+man_html_many \
+man_html_sphinx
+
+man_html_many : \
+$(DIRWWW)/manuals/html/many/index.html \
+$(LIST_SRCSH_HTML)
+
+man_html_sphinx : \
+$(DIRADM)/sphinx/conf.py \
+$(DIRTMP)/sphinx_tmpdir/index.rst \
+$(LIST_SRCSH_RST)
+ @cp $(LIST_SRCSH_RST) $(DIRTMP)/sphinx_tmpdir/
+ sphinx-build -b html -c $(DIRADM)/sphinx \
+ -d $(DIRTMP)/sphinx_tmpdir/doctrees \
+ -w $(DIRTMP)/sphinx_html.log \
+ $(DIRTMP)/sphinx_tmpdir/ \
+ $(DIRWWW)/manuals/html/sphinx/
+
+man_pdf : \
+man_pdf_many \
+man_pdf_sphinx
+
+man_pdf_many : \
+$(LIST_SRCSH_PHP)
+
+man_pdf_sphinx : \
+$(DIRADM)/sphinx/conf.py \
+$(DIRTMP)/sphinx_tmpdir/index.rst \
+$(LIST_SRCSH_RST)
+ @cp $(LIST_SRCSH_RST) $(DIRTMP)/sphinx_tmpdir/
+ @sphinx-build -b latex -c $(DIRADM)/sphinx \
+ -d $(DIRTMP)/sphinx_tmpdir/doctrees \
+ -w $(DIRTMP)/sphinx_pdf.log \
+ $(DIRTMP)/sphinx_tmpdir/ \
+ $(DIRTMP)/sphinx_tmpdir/latex_output
+ cd $(DIRTMP)/sphinx_tmpdir/latex_output/; make all-pdf
+ cp $(DIRTMP)/sphinx_tmpdir/latex_output/*.pdf \
+ $(DIRWWW)/manuals/pdf/sphinx/
+
+$(DIRWWW)/manuals/html/many/index.html : \
+$(DIRTMP)/index_many.rst
+ @$(RST2HTML) --input-encoding=ISO-8859-15 --strict \
+ $< $@
+
+$(DIRTMP)/index_many.rst :
+ @echo "$(PRODUCTNAME) manuals" | tr [:print:] = > $@
+ @echo "$(PRODUCTNAME) manuals" >> $@
+ @echo "$(PRODUCTNAME) manuals" | tr [:print:] = >> $@
+ @echo " " >> $@
+ @echo "Shell scripts" >> $@
+ @echo "Shell scripts" | tr [:print:] = >> $@
+ @for file in $(LIST_SRCSH); do echo " ";echo "$$(basename $${file})_"; echo " "; echo ".. _$$(basename $${file}) : $$(basename $${file}).html"; done >> $@
+
+$(DIRTMP)/sphinx_tmpdir/index.rst :
+ @echo ".. _index:" >> $@
+ @echo " " >> $@
+ @echo "$(PRODUCTNAME) manuals" | tr [:print:] = >> $@
+ @echo "$(PRODUCTNAME) manuals" >> $@
+ @echo "$(PRODUCTNAME) manuals" | tr [:print:] = >> $@
+ @echo " " >> $@
+ @echo "Shell scripts" >> $@
+ @echo "Shell scripts" | tr [:print:] = >> $@
+ @echo ".. toctree::" >> $@
+ @echo " :maxdepth: 1" 1>> $@
+ @echo " :glob:" 1>> $@
+ @echo " " >> $@
+ @for file in $(LIST_SRCSH); \
+ do \
+ echo " $$(basename $${file})"; \
+ done >> $@
+ @echo " " >> $@
+
+$(DIRWWW)/manuals/man/man1/%.1:$(DIRTMP)/%.rst
+ @$(RST2MAN) --input-encoding=ISO-8859-15 --strict \
+ $< $@
+
+$(DIRWWW)/manuals/html/many/%.html:$(DIRTMP)/%.rst
+ @$(RST2HTML) --input-encoding=ISO-8859-15 --strict \
+ $< $@
+
+$(DIRWWW)/manuals/pdf/many/%.pdf : $(DIRTMP)/%.tex
+ @-pdflatex -output-directory $(DIRWWW)/manuals/pdf/many/ $<
+ @-pdflatex -output-directory $(DIRWWW)/manuals/pdf/many/ $<
+
+$(DIRTMP)/%.tex : $(DIRTMP)/%.rst
+ @$(RST2LATEX) $(RST2LATEX_OPTIONS_MANY) --input-encoding=ISO-8859-15 --strict \
+ $< $@
+
+$(DIRTMP)/%.sh.rst : $(DIRSRC)/%.sh
+ @$(DIRADM)/extract_rst.sh -i $< -l sh -o $@
+
+SPECIAL_RST : $(DIRTMP)/makenemo.rst $(DIRTMP)/maketools.rst
+
+$(DIRTMP)/makenemo.rst : $(DIRSRC)/../../CONFIG/makenemo
+ @$(DIRADM)/extract_rst.sh -i $< -l sh -o $@
+$(DIRTMP)/maketools.rst : $(DIRSRC)/../maketools
+ @$(DIRADM)/extract_rst.sh -i $< -l sh -o $@
+
+
+$(DIRDESIGN)/images/%.png : $(DIRDESIGN)/images/%.svg
+ @convert $< $@
+
+$(DIRDESIGN)/images/%.svg : $(DIRDESIGN)/%.dot
+ @dot -Tsvg -o $@ $<
+
+$(DIRDESIGN)/$(PRODUCT)_fulldependencies.dot : \
+./makefile
+ @makeppgraph --graphviz --output=$@
Index: /branches/UKMO/dev_r5518_GO6_under_ice_relax_dr_hook/ADM/DOC_SCRIPTS/sphinx/conf.py
===================================================================
--- /branches/UKMO/dev_r5518_GO6_under_ice_relax_dr_hook/ADM/DOC_SCRIPTS/sphinx/conf.py (revision 11643)
+++ /branches/UKMO/dev_r5518_GO6_under_ice_relax_dr_hook/ADM/DOC_SCRIPTS/sphinx/conf.py (revision 11643)
@@ -0,0 +1,202 @@
+# -*- coding: utf-8 -*-
+#
+# superbib documentation build configuration file, created by
+# sphinx-quickstart on Wed Mar 10 16:19:33 2010.
+#
+# This file is execfile()d with the current directory set to its containing dir.
+#
+# Note that not all possible configuration values are present in this
+# autogenerated file.
+#
+# All configuration values have a default; values that are commented out
+# serve to show the default.
+
+import sys, os
+
+# If extensions (or modules to document with autodoc) are in another directory,
+# add these directories to sys.path here. If the directory is relative to the
+# documentation root, use os.path.abspath to make it absolute, like shown here.
+#sys.path.append(os.path.abspath('/usr/home/fplod/src/superbib_ws/adm/sphinx_tmpdir/source/'))
+
+
+# -- General configuration -----------------------------------------------------
+
+# Add any Sphinx extension module names here, as strings. They can be extensions
+# coming with Sphinx (named 'sphinx.ext.*') or your custom ones.
+extensions = ['sphinx.ext.todo']
+todo_include_todos = True
+
+
+# Add any paths that contain templates here, relative to this directory.
+templates_path = ['_templates']
+
+# The suffix of source filenames.
+source_suffix = '.rst'
+
+# The encoding of source files.
+source_encoding = 'iso-8859-15'
+
+# The master toctree document.
+master_doc = 'index'
+
+# General information about the project.
+project = u'NEMO FCM'
+copyright = u'2010, CNRS'
+
+# The version info for the project you're documenting, acts as replacement for
+# |version| and |release|, also used in various other places throughout the
+# built documents.
+#
+# The short X.Y version.
+version = '1.0.1'
+# The full version, including alpha/beta/rc tags.
+release = '1.0.1'
+
+# The language for content autogenerated by Sphinx. Refer to documentation
+# for a list of supported languages.
+language = 'en'
+
+# There are two options for replacing |today|: either, you set today to some
+# non-false value, then it is used:
+#today = ''
+# Else, today_fmt is used as the format for a strftime call.
+#today_fmt = '%B %d, %Y'
+today_fmt = '%Y%m%d'
+
+# List of documents that shouldn't be included in the build.
+#unused_docs = []
+
+# List of directories, relative to source directory, that shouldn't be searched
+# for source files.
+exclude_trees = []
+
+exclude_dirnames = ['.svn']
+
+# The reST default role (used for this markup: `text`) to use for all documents.
+#default_role = None
+
+# If true, '()' will be appended to :func: etc. cross-reference text.
+#add_function_parentheses = True
+
+# If true, the current module name will be prepended to all description
+# unit titles (such as .. function::).
+#add_module_names = True
+
+# If true, sectionauthor and moduleauthor directives will be shown in the
+# output. They are ignored by default.
+#show_authors = False
+
+# The name of the Pygments (syntax highlighting) style to use.
+pygments_style = 'sphinx'
+
+# A list of ignored prefixes for module index sorting.
+modindex_common_prefix = []
+
+
+# -- Options for HTML output ---------------------------------------------------
+
+# The theme to use for HTML and HTML Help pages. Major themes that come with
+# Sphinx are currently 'default' and 'sphinxdoc'.
+html_theme = 'default'
+
+# Theme options are theme-specific and customize the look and feel of a theme
+# further. For a list of options available for each theme, see the
+# documentation.
+#html_theme_options = {}
+
+# Add any paths that contain custom themes here, relative to this directory.
+#html_theme_path = []
+
+# The name for this set of Sphinx documents. If None, it defaults to
+# " v documentation".
+#html_title = None
+
+# A shorter title for the navigation bar. Default is the same as html_title.
+#html_short_title = None
+
+# The name of an image file (relative to this directory) to place at the top
+# of the sidebar.
+#html_logo = None
+
+# The name of an image file (within the static path) to use as favicon of the
+# docs. This file should be a Windows icon file (.ico) being 16x16 or 32x32
+# pixels large.
+#html_favicon = None
+
+# Add any paths that contain custom static files (such as style sheets) here,
+# relative to this directory. They are copied after the builtin static files,
+# so a file named "default.css" will overwrite the builtin "default.css".
+html_static_path = ['_static']
+
+# If not '', a 'Last updated on:' timestamp is inserted at every page bottom,
+# using the given strftime format.
+#html_last_updated_fmt = '%b %d, %Y'
+
+# If true, SmartyPants will be used to convert quotes and dashes to
+# typographically correct entities.
+#html_use_smartypants = True
+
+# Custom sidebar templates, maps document names to template names.
+#html_sidebars = {}
+
+# Additional templates that should be rendered to pages, maps page names to
+# template names.
+#html_additional_pages = {}
+
+# If false, no module index is generated.
+html_use_modindex = True
+
+# If false, no index is generated.
+html_use_index = True
+
+# If true, the index is split into individual pages for each letter.
+#html_split_index = False
+
+# If true, links to the reST sources are added to the pages.
+html_show_sourcelink = False
+
+# If true, an OpenSearch description file will be output, and all pages will
+# contain a tag referring to it. The value of this option must be the
+# base URL from which the finished HTML is served.
+#html_use_opensearch = ''
+
+# If nonempty, this is the file name suffix for HTML files (e.g. ".xhtml").
+#html_file_suffix = ''
+
+# Output file base name for HTML help builder.
+htmlhelp_basename = 'SETTE validation tool'
+
+
+# -- Options for LaTeX output --------------------------------------------------
+
+# The paper size ('letter' or 'a4').
+#latex_paper_size = 'letter'
+
+# The font size ('10pt', '11pt' or '12pt').
+#latex_font_size = '10pt'
+
+# Grouping the document tree into LaTeX files. List of tuples
+# (source start file, target name, title, author, documentclass [howto/manual]).
+latex_documents = [
+ ('index', 'sette_tools.tex', u'NEMO validation tools Documentation',
+ u'C. Levy', 'manual'),
+]
+
+# The name of an image file (relative to this directory) to place at the top of
+# the title page.
+#latex_logo = None
+
+# For "manual" documents, if this is true, then toplevel headings are parts,
+# not chapters.
+#latex_use_parts = False
+
+# Additional stuff for the LaTeX preamble.
+#latex_preamble = ''
+
+# Documents to append as an appendix to all manuals.
+#latex_appendices = []
+
+# If false, no module index is generated.
+#latex_use_modindex = True
+
+keep_warnings = 'True'
Index: /branches/UKMO/dev_r5518_GO6_under_ice_relax_dr_hook/ADM/README
===================================================================
--- /branches/UKMO/dev_r5518_GO6_under_ice_relax_dr_hook/ADM/README (revision 11643)
+++ /branches/UKMO/dev_r5518_GO6_under_ice_relax_dr_hook/ADM/README (revision 11643)
@@ -0,0 +1,9 @@
+Generates automatic documentation: in ADM/DOC_SCRIPTS
+Pre requisite: to have sphinx installed
+Input file is in ADM/DOC_SCRIPTS/sphinx/conf.py to be checked/modifies by user
+
+Once the input file is set up, run:
+cd ADM/DOC_SCRIPTS
+gmake -f makefile_compile before
+gmake -f makefile_compile man_pdf
+to generate documentation in pdf in ADM/DOC_SCRIPTS/doc/manuals/pdf/sphinx
Index: /branches/UKMO/dev_r5518_GO6_under_ice_relax_dr_hook/DOC/NEMO_book.tex
===================================================================
--- /branches/UKMO/dev_r5518_GO6_under_ice_relax_dr_hook/DOC/NEMO_book.tex (revision 11643)
+++ /branches/UKMO/dev_r5518_GO6_under_ice_relax_dr_hook/DOC/NEMO_book.tex (revision 11643)
@@ -0,0 +1,340 @@
+%description: Book template
+
+% template of document for LaTeX
+% (C) Xavier Perseguers 2002 - xavier.perseguers@epfl.ch
+
+\documentclass[a4paper,11pt]{book}
+%\documentclass[a4paper,11pt,makeidx]{book} <== may need this to generate index
+
+% makeindex NEMO_book <== to regenerate the index
+% bibtex NEMO_book <== to generate the bibliography
+
+% ================================================================
+% HEADERS DEFINITION
+% ================================================================
+
+\usepackage[french]{babel}
+%\usepackage{color}
+\usepackage{xcolor}
+%\usepackage{graphics} % allows insertion of pictures
+\usepackage{graphicx} % allows insertion of pictures
+\usepackage[capbesideposition={top,center}]{floatrow} % allows captions
+\floatsetup[table]{style=plaintop} % beside pictures
+\usepackage[margin=10pt,font={small},labelsep=colon,labelfont={bf}]{caption} % Gives small font for captions
+\usepackage{enumitem} % allows non-bold description items
+\usepackage{longtable} % allows multipage tables
+%\usepackage{colortbl} % gives coloured panels behind table columns
+
+%hyperref
+\usepackage[ %
+ pdftitle={NEMO ocean engine}, %
+ pdfauthor={Gurvan Madec}, % pdfsubject={The preprint document class
+ % elsart},% pdfkeywords={diapycnal diffusion,numerical mixing,z-level models},%
+ pdfstartview=FitH, %
+ bookmarks=true, %
+ bookmarksopen=true, %
+ breaklinks=true, %
+ colorlinks=true, %
+ linkcolor=blue,anchorcolor=blue, %
+ citecolor=blue,filecolor=blue, %
+ menucolor=blue, %
+ urlcolor=blue]{hyperref}
+% usage of exteranl hyperlink : \href{mailto:my_address@wikibooks.org}{my\_address@wikibooks.org}
+% \url{http://www.wikibooks.org}
+% or \href{http://www.wikibooks.org}{wikibooks home}
+
+
+
+%%%% page styles etc................
+\usepackage{fancyhdr}
+\pagestyle{fancy}
+% with this we ensure that the chapter and section
+% headings are in lowercase.
+\renewcommand{\chaptermark}[1]{\markboth{#1}{}}
+\renewcommand{\sectionmark}[1]{\markright{\thesection.\ #1}}
+\fancyhf{} % delete current setting for header and footer
+\fancyhead[LE,RO]{\bfseries\thepage}
+\fancyhead[LO]{\bfseries\hspace{-0em}\rightmark}
+\fancyhead[RE]{\bfseries\leftmark}
+\renewcommand{\headrulewidth}{0.5pt}
+\renewcommand{\footrulewidth}{0pt}
+\addtolength{\headheight}{2.6pt} % make space for the rule
+%\addtolength{\headheight}{1.6pt} % make space for the rule
+\fancypagestyle{plain}{
+ \fancyhead{} % get rid of headers on plain pages
+ \renewcommand{\headrulewidth}{0pt} % and the line
+}
+
+
+%%%% Section number in Margin.......
+% typeset the number of each section in the left margin, with the start of each instance of
+% sectional heading text aligned with the left hand edge of the body text.
+\makeatletter
+\def\@seccntformat#1{\protect\makebox[0pt][r]{\csname the#1\endcsname\quad}}
+\makeatother
+
+% Leave blank pages completely empty, w/o header
+\makeatletter
+\def\cleardoublepage{\clearpage\if@twoside \ifodd\c@page\else
+ \hbox{}
+ \vspace*{\fill}
+ \vspace{\fill}
+ \thispagestyle{empty}
+ \newpage
+ \if@twocolumn\hbox{}\newpage\fi\fi\fi}
+\makeatother
+
+%%%% define the chapter style ................
+\usepackage{minitoc} %In French : \usepackage[french]{minitoc}
+%\usepackage{mtcoff} % invalidate the use of minitocs
+\usepackage{fancybox}
+
+\makeatletter
+\def\LigneVerticale{\vrule height 5cm depth 2cm\hspace{0.1cm}\relax}
+\def\LignesVerticales{%
+ \let\LV\LigneVerticale\LV\LV\LV\LV\LV\LV\LV\LV\LV\LV}
+\def\GrosCarreAvecUnChiffre#1{%
+ \rlap{\vrule height 0.8cm width 1cm depth 0.2cm}%
+ \rlap{\hbox to 1cm{\hss\mbox{\color{white} #1}\hss}}%
+ \vrule height 0pt width 1cm depth 0pt}
+\def\GrosCarreAvecTroisChiffre#1{%
+ \rlap{\vrule height 0.8cm width 1.6cm depth 0.2cm}%
+ \rlap{\hbox to 1.5cm{\hss\mbox{\color{white} #1}\hss}}%
+ \vrule height 0pt width 1cm depth 0pt}
+
+\def\@makechapterhead#1{\hbox{%
+ \huge
+ \LignesVerticales
+ \hspace{-0.5cm}%
+ \GrosCarreAvecUnChiffre{\thechapter}
+ \hspace{0.2cm}\hbox{#1}%
+% \GrosCarreAvecTroisChiffre{\thechapter}
+% \hspace{1cm}\hbox{#1}%
+%}\par\vskip 2cm}
+}\par\vskip 1cm}
+\def\@makeschapterhead#1{\hbox{%
+ \huge
+ \LignesVerticales
+ %\hspace{0.5cm}%
+ \hbox{#1}%
+}\par\vskip 2cm}
+\makeatother
+
+%\def\thechapter{\Roman{chapter}} % chapter number to be Roman
+
+
+%%%% Mathematics...............
+%\documentclass{amsart}
+\usepackage{xspace} % helpd ensure correct spacing after macros
+\usepackage{latexsym}
+\usepackage{amssymb}
+\usepackage{amsmath}
+\allowdisplaybreaks[1] % allow page breaks in the middle of equations
+\usepackage{./TexFiles/math_abbrev} % use maths shortcuts
+
+
+\usepackage{times} % use times font for text
+%\usepackage{mathtime} % font for illustrator to work (belleek fonts )
+%\usepackage[latin1]{inputenc} % allows some unicode removed (agn)
+
+
+%%% essai commande
+\newcommand{\nl} [1] {\texttt{\small {\textcolor{blue}{#1}} } }
+\newcommand{\nlv} [1] {\texttt{\footnotesize#1}\xspace}
+\newcommand{\smnlv} [1] {\texttt{\scriptsize#1}\xspace}
+
+%%%% namelist & code display................................
+\usepackage{alltt} %% alltt for namelist
+\usepackage{verbatim} %% alltt for namelist
+% namelists
+\newcommand{\namdisplay} [1] {
+\begin{alltt}
+{\tiny \verbatiminput{./TexFiles/Namelist/#1}}
+\end{alltt}
+ \vspace{-10pt}
+}
+% code display
+%\newcommand{\codedisplay} [1] { \begin{alltt} {\tiny {\begin{verbatim} {#1}} \end{verbatim} } \end{alltt} }
+
+
+
+%%%% commands for working with text................................
+% command to "comment out" portions of text ({} argument) or not ({#1} argument)
+\newcommand{\amtcomment}[1]{} % command to "commented out" portions of text or not (#1 in argument)
+\newcommand{\sgacomment}[1]{} % command to "commented out" portions of
+\newcommand{\gmcomment}[1]{} % command to "commented out" portions of
+% % text that span line breaks
+%Red (NR) or Yellow(WARN)
+%\newcommand{\NR} {\colorbox{red}{#1}}
+%\newcommand{\WARN} {{ \colorbox{yellow}{#1}} }
+
+
+
+%%% index commands......................
+\usepackage{makeidx}
+%\usepackage{showidx} % show the index entry
+
+\newcommand{\mdl} [1] {\textit{#1.F90}\index{Modules!#1}} %module (mdl)
+\newcommand{\rou} [1] {\textit{#1}\index{Routines!#1}} %module (routine)
+\newcommand{\hf} [1] {\textit{#1.h90}\index{h90 file!#1}} %module (h90 files)
+\newcommand{\ngn} [1] {\textit{#1}\index{Namelist Group Name!#1}} %namelist name (nampar)
+\newcommand{\np} [1] {\textit{#1}\index{Namelist variables!#1}} %namelist variable
+\newcommand{\jp} [1] {\textit{#1}\index{Model parameters!#1}} %model parameter (jp)
+\newcommand{\pp} [1] {\textit{#1}\index{Model parameters!#1}} %namelist parameter (pp)
+\newcommand{\ifile} [1] {\textit{#1.nc}\index{Input NetCDF files!#1.nc}} %input NetCDF files (.nc)
+\newcommand{\key} [1] {\textbf{key\_#1}\index{CPP keys!key\_#1}} %key_cpp (key)
+\newcommand{\NEMO} {\textit{NEMO}\xspace} %NEMO (nemo)
+
+%%%% Bibliography .............
+\usepackage[nottoc, notlof, notlot]{tocbibind}
+\usepackage[square, comma]{natbib}
+\bibpunct{[}{]}{,}{a}{}{;} %suppress "," after "et al."
+\providecommand{\bibfont}{\small}
+
+
+% ================================================================
+% FRONT PAGE
+% ================================================================
+
+%\usepackage{pstricks}
+\title{
+%\psset{unit=1.1in,linewidth=4pt} %parameters of the units for pstricks
+% \rput(0,2){ \includegraphics[width=1.1\textwidth]{./TexFiles/Figures/logo_ALL.pdf} } \\
+% \vspace{0.1cm}
+\vspace{-6.0cm}
+\includegraphics[width=1.1\textwidth]{./TexFiles/Figures/logo_ALL.pdf}\\
+\vspace{5.1cm}
+\includegraphics[width=0.9\textwidth]{./TexFiles/Figures/NEMO_logo_Black.pdf} \\
+\vspace{1.4cm}
+\rule{345pt}{1.5pt} \\
+\vspace{0.45cm}
+{\Huge NEMO ocean engine}
+\rule{345pt}{1.5pt} \\
+ }
+%{ -- Draft --} }
+%\date{\today}
+\date{
+November 2014 \\
+{\small -- version 3.6 --} \\
+~ \\
+\textit{\small Note du P\^ole de mod\'{e}lisation de l'Institut Pierre-Simon Laplace No 27 }\\
+\vspace{0.45cm}
+{ ISSN No 1288-1619.}
+}
+
+
+\author{
+\Large Gurvan Madec, and the NEMO team \\
+ \texttt{\small gurvan.madec@locean-ipsl.umpc.fr} \\
+ \texttt{\small nemo\_st@locean-ipsl.umpc.fr} \\
+%{\small Laboratoire d'Oc\'{e}anographie et du Climat: Exp\'{e}rimentation et Approches Num\'{e}riques }
+}
+
+\dominitoc
+\makeindex %type this first : makeindex -s NEMO.ist NEMO_book.idx
+
+% ================================================================
+% Include ONLY order
+% ================================================================
+
+%\includeonly{./TexFiles/Chapters/Chap_MISC}
+%\includeonly{./TexFiles/Chapters/Chap_ZDF}
+%\includeonly{./TexFiles/Chapters/Chap_STP,./TexFiles/Chapters/Chap_SBC,./TexFiles/Chapters/Chap_TRA}
+%\includeonly{./TexFiles/Chapters/Chap_LBC,./TexFiles/Chapters/Chap_MISC}
+%\includeonly{./TexFiles/Chapters/Chap_Model_Basics}
+%\includeonly{./TexFiles/Chapters/Annex_A,./TexFiles/Chapters/Annex_B,./TexFiles/Chapters/Annex_C,./TexFiles/Chapters/Annex_D}
+
+% ================================================================
+% ================================================================
+
+\begin{document}
+
+\maketitle % generate the title
+
+\frontmatter
+
+\tableofcontents % generate a table of contents
+%\listoffigures % generate a list of figures
+%\listoftables % generate a list of tables
+
+\mainmatter
+
+% ================================================================
+% Abstract - Foreword
+% ================================================================
+
+\include{./TexFiles/Chapters/Abstracts_Foreword}
+
+% ================================================================
+% INTRODUCTION
+% ================================================================
+
+\include{./TexFiles/Chapters/Introduction}
+
+% ================================================================
+% CHAPTERS
+% ================================================================
+
+\include{./TexFiles/Chapters/Chap_Model_Basics}
+
+\include{./TexFiles/Chapters/Chap_STP} % Time discretisation (time stepping strategy)
+
+\include{./TexFiles/Chapters/Chap_DOM} % Space discretisation
+
+\include{./TexFiles/Chapters/Chap_TRA} % Tracer advection/diffusion equation
+
+\include{./TexFiles/Chapters/Chap_DYN} % Dynamics : momentum equation
+
+\include{./TexFiles/Chapters/Chap_SBC} % Surface Boundary Conditions
+
+\include{./TexFiles/Chapters/Chap_LBC} % Lateral Boundary Conditions
+
+\include{./TexFiles/Chapters/Chap_LDF} % Lateral diffusion
+
+\include{./TexFiles/Chapters/Chap_ZDF} % Vertical diffusion
+
+\include{./TexFiles/Chapters/Chap_DIA} % Outputs and Diagnostics
+
+\include{./TexFiles/Chapters/Chap_OBS} % Observation operator
+
+\include{./TexFiles/Chapters/Chap_ASM} % Assimilation increments
+
+\include{./TexFiles/Chapters/Chap_STO} % Stochastic param.
+
+\include{./TexFiles/Chapters/Chap_MISC} % Miscellaneous topics
+
+\include{./TexFiles/Chapters/Chap_CFG} % Predefined configurations
+
+% ================================================================
+% APPENDIX
+% ================================================================
+
+\appendix
+
+%\include{./TexFiles/Chapters/Chap_Conservation}
+\include{./TexFiles/Chapters/Annex_A} % generalised vertical coordinate
+\include{./TexFiles/Chapters/Annex_B} % diffusive operator
+\include{./TexFiles/Chapters/Annex_C} % Discrete invariants of the eqs.
+\include{./TexFiles/Chapters/Annex_D} % Coding rules
+\include{./TexFiles/Chapters/Annex_ISO} % Isoneutral diffusion using triads
+%\include{./TexFiles/Chapters/Annex_E} % Notes on some on going staff (no included in the DOC)
+%\include{./TexFiles/Chapters/Annex_Fox-Kemper} % Notes on Fox-Kemper (no included in the DOC)
+%\include{./TexFiles/Chapters/Annex_EVP} % Notes on EVP (no included in the DOC)
+
+% ================================================================
+% INDEX
+% ================================================================
+
+\addcontentsline{toc}{chapter}{Index}
+\printindex
+
+% ================================================================
+% BIBLIOGRAPHY
+% ================================================================
+
+%%\bibliographystyle{plainat}
+\bibliographystyle{./TexFiles/ametsoc} % AMS biblio style (JPO)
+\bibliography{./TexFiles/Biblio/Biblio}
+
+% ================================================================
+\end{document}
Index: /branches/UKMO/dev_r5518_GO6_under_ice_relax_dr_hook/DOC/NEMO_coding.conv.tex
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+\documentclass[a4paper]{article}
+\usepackage{type1cm}
+\usepackage{times}
+\usepackage{color}
+\usepackage{rotating}
+\usepackage{color}
+\usepackage{framed}
+\usepackage{makeidx}
+
+
+%%%%%%%
+\pagestyle{empty}
+\setlength{\leftmargin}{1 cm}
+\setlength{\rightmargin}{1 cm}
+\setlength{\oddsidemargin}{0 cm}
+\setlength{\evensidemargin}{0 cm}
+\setlength{\topmargin}{-1cm}
+\setlength{\textwidth}{16 cm}
+\setlength{\textheight}{25cm}
+
+%%%%%%%%%essai plus jolis from NEMO book
+\usepackage{fancyhdr}
+
+\pagestyle{fancy}
+%\usepackage[colorlinks=true]{hyperref} %%create link
+
+\makeindex %% run first makeindex NEMO_coding.conv.idx NEMO_coding.conv.ist
+
+\begin{document}
+
+
+\title{
+\includegraphics[width=0.3\textwidth]{./TexFiles/Figures/NEMO_logo_Black.pdf} \\
+\vspace{1.0cm}
+\rule{345pt}{1.5pt} \\
+\vspace{0.45cm}
+ {\Huge NEMO coding conventions}
+\rule{345pt}{1.5pt} \\
+{\small -- version 3 --} }
+%\title{NEMO coding conventions}
+\author{NEMO System Team }
+\date{March 2011}
+
+
+\maketitle
+
+\newpage
+
+\tableofcontents
+
+\newpage
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+\section{Introduction}
+This document describes conventions\index{conventions} used in NEMO coding and suggested for its development. The objectives are to offer a guide to all readers of the NEMO code, and to facilitate the work of all the developers, including the validation of their developments, and eventually the implementation of these developments within the NEMO platform. \\
+A first approach of these rules can be found in the code in $NEMO/OPA\_SRC/module\_example$ where all the basics coding conventions are illustrated. More details can be found below.\\
+This work is based on the coding conventions in use for the Community Climate System Model, \footnote { http://www.cesm.ucar.edu/working\_groups/Software/dev\_guide/dev\_guide/node7.html }
+ the previous version of this document (``FORTRAN coding standard in the OPA System'') and the expertise of the NEMO System Team which can be contacted for further information ($nemo\_st@locean-ipsl.upmc.fr$)
+After a general overview below, this document will describe :
+\begin{itemize}
+\item The style rules, i.e. the syntax, appearance and naming conventions chosen to improve readability of the code;
+\item The content rules, i.e. the conventions to improve the reliability of the different parts of the code;
+\item The package rules to go a step further by improving the reliability of the whole and interfaces between routines and modules.
+\end{itemize}
+
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+\section{Overview and general conventions}
+NEMO has several different components: ocean dynamics ($OPA\_SRC$), sea-ice ($LIM\_SRC$), ocean biogeochemistry\- ($TOP\_SRC$), linear-tangent and adjoint of the dynamics ($TAM$)É each of them corresponding to a directory.
+In each directory, one will find some FORTRAN files and/or subdirectories, one per functionality of the code: $BDY$ (boundaries), $DIA$ (diagnostics), $DOM$ (domain), $DYN$ (dynamics), $LDF$ (lateral diffusion), etc...\\
+All name are chosen to be as self-explanatory as possible, in English, all prefixes are 3 digits.\\
+English is used for all variables names, comments, and documentation. \\
+Physical units are MKS. The only exception to this is the temperature, which is expressed in degrees Celsius, except in bulk formulae and part of LIM sea-ice model where it is in Kelvin. See $DOM/phycst.F90$ files for conversions.
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+\section{Architecture}
+Within each directory, organisation of files is driven by ÒorthogonalityÓ\index{orthogonality}, i.e. one functionality of the code is intended to be in one and only one directory, and one module and all its related routines are in one file.
+The functional modules\index{module} are:
+\begin{itemize}
+\item SBC surface module
+\item IOM management of the I/O
+\item NST interface to AGRIF (nesting model) for dynamics and biogeochemistry
+\item OBC, BDY management of structured and unstructured open boundaries
+\item C1D 1D (vertical) configuration for dynamics, sea-ice and biogeochemistry
+\item OFF off-line module: passive tracer or biogeochemistry alone
+\item CFG tutorial and reference configurations
+\item DOC documentation
+\end{itemize}
+
+For example, the file $domain.F90$ contains the module $domain$ and all the subroutines related to this module ($ dom\_init, dom\_nam, dom\_ctl$).
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+\section{Style rules}
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+\subsection{Argument list format}
+Routine argument lists will contain a maximum 5 variables\index{variable} per line, whilst continuation lines can be used.
+This applies both to the calling routine and the dummy argument list in the routine being called. The purpose is to simplify matching up the arguments between caller and callee.
+
+\begin{verbatim}
+SUBROUTINE tra_adv_eiv( kt, pun, pvn, pwn )
+
+ CALL tra_adv_eiv( kt, zun, zvn, zwn )
+\end{verbatim}
+
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+\subsection{Array syntax}
+Except for long loops (see below), array notation should be used if possible. To improve readability the array shape must be shown in brackets, e.g.:
+\begin{verbatim}
+onedarraya(:) = onedarrayb(:) + onedarrayc(:)
+twodarray (:,:) = scalar * anothertwodarray(:,:)
+\end{verbatim}
+When accessing sections of arrays, for example in finite difference equations, do so by using the triplet notation on the full array, e.g.:
+\begin{verbatim}
+twodarray(:,2:len2) = scalar &
+ & * ( twodarray2(:,1:len2-1 ) &
+ & - twodarray2(:,2:len2 ) )
+\end{verbatim}
+For long, complicated loops, explicitly indexed loops should be preferred. In general when using this syntax, the order of the loops indices should reflect the following scheme (for best usage of data locality):
+\begin{verbatim}
+DO jk = 1, jpk
+ DO jj = 1, jpj
+ DO ji = 1, jpi
+ array(ji,jj,jk) = ...
+ END DO
+ END DO
+END DO
+\end{verbatim}
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+\subsection{Case}
+All FORTRAN keywords are in capital : \begin {verbatim} DIMENSION, WRITE, DO, END DO, NAMELIST \end{verbatim}
+All other parts of the NEMO code will be written in lower case.
+
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+\subsection{Comments}
+Comments in the code are useful when reading the code and changing or developing it. \\
+The full documentation and detailed explanations are to be added in the reference manual (TeX files, aside from the code itself). \\
+In the code, the comments should explain variable content and describe each computational step.\\
+Comments in the header start with ``!!''. For more details on the content of the headers, see ÒContent rules/HeadersÓ in this document.\\
+Comments in the code start with ``!''.\\
+All comments are indented (3, 6, or 9 É blank spaces).\\
+Short comments may be included on the same line as executable code, and an additional line can be used with proper alignment. For example:
+\begin{verbatim}
+ zx = zx *zzy ! Describe what is going on and if it is
+ ! ! too long use another Ô!Õ for proper
+ ! ! alignment with automatic indentation
+\end{verbatim}
+More in-depth comments should be written in the form:
+\begin{verbatim}
+! Check of some namelist values
+\end{verbatim}
+or
+\begin{verbatim}
+!
+! !<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<
+! ! Bottom boundary condition on tke
+! !<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<
+!
+\end{verbatim}
+Key features of this style are 1) it starts with a "!" in the column required for proper indentation, 2) the text is offset above and below by a blank line or a content line built for underlying.
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+\subsection{Continuation lines}
+Continuation lines can be used with precise alignment for readability. For example:
+\begin{verbatim}
+avmu(ji,jj,jk) = avmu(ji,jj,jk) * ( un(ji,jj,jk-1) - un(ji,jj,jk) ) &
+ & * ( ub(ji,jj,jk-1) - ub(ji,jj,jk) ) &
+ & / ( fse3uw_n(ji,jj,jk) &
+ & * fse3uw_b(ji,jj,jk) )
+\end{verbatim}
+Code lines, which are continuation lines of assignment statements, must begin to the right of the column of the assignment operator. Due to the possibility of automatic indentation in some editor (emacs for example), use a ``\&'' as first character of the continuing lines to maintain the alignment.
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+\subsection{Declaration of arguments and local variables}
+
+In a routine, input arguments and local variables are declared 1 per line, with a comment field on the same line as the declaration. Multiple comment lines describing a single variable are acceptable if needed. For example:
+\begin{verbatim}
+INTEGER :: kstp ! ocean time-step index
+\end{verbatim}
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+\subsection{F90 Standard}
+NEMO software adheres to the FORTRAN 95 language standard and does not rely on any specific language or vendor extensions.
+
+
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+\subsection{Free-Form Source}
+Free-form source will be used. The F90/95 standard allows lines of up to 132 characters, but a self-imposed limit of 80 should enhance readability, or print source files with two columns per page. Multi-line comments that extend to column 100 are unacceptable.
+
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+\subsection{Indentation}
+Code as well as comment lines within loops, if-blocks, continuation lines, MODULE or SUBROUTINE statements will be indented 3 characters for readability. (except for CONTAINS that remains at first column)
+\begin{verbatim}
+MODULE mod1
+ REAL(wp) xx
+CONTAINS
+ SUBROUTINE sub76( px, py, pz, pw, pa, &
+ & pb, pc, pd, pe )
+
+ END SUBROUTINE sub76
+END MODULE mod1
+\end{verbatim}
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+\subsection{Loops}
+Loops, if explicit, should be structured with the do-end do construct as opposed to numbered loops. Nevertheless non-numeric labels can be used for a big iterative loop of a recursive algorithm. In the case of a long loop, a self-descriptive label can be used (i.e. not just a number).
+
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+\subsection{Naming Conventions: files}
+A file containing a module will have the same name as the module it contains (because dependency rules used by "make" programs are based on file names).
+\footnote{For example, if routine A "USE"s module B, then "make" must be told of the dependency relation which requires B to be compiled before A. If one can assume that module B resides in file B.o, building a tool to generate this dependency rule (e.g. A.o: B.o) is quite simple. Put another way, it is difficult (to say nothing of CPU-intensive) to search an entire source tree to find the file in which module B resides for each routine or module which "USE"s B.}
+
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+\subsection{Naming Conventions: modules}
+Use a meaningful English name and the ``3 letters'' naming convention: first 3 letters for the code section, and last 3 to describe the module. For example, zdftke, where ``zdf'' stands for vertical diffusion, and ``tke'' for turbulent kinetic energy.
+\\
+Note that by implication multiple modules are not allowed in a single file.
+The use of common blocks is deprecated in Fortran 90 and their use in NEMO is strongly discouraged. Modules are a better way to declare static data. Among the advantages of modules is the ability to freely mix data of various types, and to limit access to contained variables through the use of the ONLY and PRIVATE attributes.
+
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+\subsection{Naming Conventions: variables}
+All variable should be named as explicitly as possible in English. The naming convention concerns prefix letters of these name, in order to identify the variable type and status.\\
+Never use a FORTRAN keyword as a routine or variable name. \\
+The table below lists the starting letter(s) to be used for variable naming, depending on their type and status:
+%--------------------------------------------------TABLE--------------------------------------------------
+\begin{table}[htbp]
+\begin{center}
+\begin{tabular}{|p{50pt}|p{50pt}|p{50pt}|p{50pt}|p{50pt}|p{50pt}|p{50pt}|}
+\hline Type \par / Status & integer& real& logical & character& double \par precision& complex \\
+\hline
+public \par or \par module variable&
+\textbf{m n} \par \textit{but not } \par \textbf{nn\_}&
+\textbf{a b e f g h o} \textbf{q} \textit{to} \textbf{x} \par but not \par \textbf{fs rn\_}&
+\textbf{l} \par \textit{but not} \par \textbf{lp ld ll ln\_}&
+\textbf{c} \par \textit{but not} \par \textbf{cp cd cl cn\_}&
+\textbf{d} \par \textit{but not} \par \textbf{dp dd dl dn\_}&
+\textbf{y} \par \textit{but not} \par \textbf{yp yd yl} \\
+\hline
+dummy \par argument&
+\textbf{k} \par \textit{but not} \par \textbf{kf}&
+\textbf{p} \par \textit{but not} \par \textbf{pp pf}&
+\textbf{ld}&
+\textbf{cd}&
+\textbf{dd}&
+\textbf{yd} \\
+\hline
+local \par variable&
+\textbf{i}&
+\textbf{z}&
+\textbf{ll}&
+\textbf{cl}&
+\textbf{cd}&
+\textbf{yl} \\
+\hline
+loop \par control&
+\textbf{j} \par \textit{but not } \par \textbf{jp}&
+&
+&
+&
+&
+ \\
+\hline
+parameter&
+\textbf{jp}&
+\textbf{pp}&
+\textbf{lp}&
+\textbf{cp}&
+\textbf{dp}&
+\textbf{yp} \\
+\hline
+
+namelist&
+\textbf{nn\_}&
+\textbf{rn\_}&
+\textbf{ln\_}&
+\textbf{cn\_}&
+\textbf{dn\_}&
+\\
+\hline
+CPP \par macro&
+\textbf{kf}&
+\textbf{sf} \par &
+&
+&
+&
+ \\
+\hline
+\end{tabular}
+\label{tab1}
+\end{center}
+\end{table}
+%--------------------------------------------------------------------------------------------------------------
+
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+\subsection{Operators}
+Use of the operators $<, >, <=, >=, ==, /= $ is strongly recommended instead of their deprecated counterparts, $lt., .gt., .le., .ge., .eq., and .ne. $ The motivation is readability. In general use the notation: \\
+$$
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+\subsection{Pre processor}
+Where the use of a language pre-processor is required, it will be the C pre-processor (cpp).\\
+The cpp key is the main feature used, allowing to ignore some useless parts of the code at compilation step. \\
+The advantage is to reduce the memory use; the drawback is that compilation of this part of the code isn't checked. \\
+The cpp key feature should only be used for a few limited options, if it reduces the memory usage. In all cases, a logical variable and a FORTRAN $IF$ should be preferred.
+When using a cpp key $key\_optionname$, a corresponding logical variable $lk\_optionname$ should be declared to allow FORTRAN $IF$ tests in the code and a FORTRAN module with the same name (i.e. $optionname.F90$) should
+ be defined. This module is the only place where a ``\#if defined'' command appears, selecting either the whole FORTRAN code or a dummy module. For example, the TKE vertical physics, the module name is $zdftke.F90$, the CPP key is $key\_zdftke$ and the associated logical is $lk\_zdftke$.
+
+The following syntax:
+\begin{verbatim}
+#if defined key_optionname
+!! Part of code conditionally compiled if cpp key key_optionname is active
+#endif
+\end{verbatim}
+Is to be used rather than the \#ifdef abbreviate form since it may have conflicts with some Unix scripts.
+
+Tests on cpp keys included in NEMO at compilation step:
+\begin{itemize}
+\item The CPP keys used are compared to the previous list of cpp keys (the compilation will stop if trying to specify a Ònon-existing keyÓ)
+\item If a change occurs in the CPP keys used for a given experiment, the whole compilation phase is done again.
+\end{itemize}
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+\section{Content rules}
+
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+\subsection{Configurations}
+The configuration defines the domain and the grid on which NEMO is running. It may be useful to associate a cpp key and some variables to a given configuration, although the part of the code changed under each of those keys should be minimized. As an example, the "ORCA2" configuration (global ocean, 2 degrees grid size) is associated with the cpp key $key\_orca2$ for which
+\begin{verbatim}
+cp_cfg = "orca"
+jp_cfg = 2
+\end{verbatim}
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+\subsection{Constants}
+Physical constants (e.g. pi, gas constants) must never be hardwired into the executable portion of a code. Instead, a mnemonically named variable or parameter should be set to the appropriate value, in the setup routine for the package\index{package}. We realize than many parameterizations rely on empirically derived constants or fudge factors, which are not easy to name. In these cases it is not forbidden to leave such factors coded as "magic numbers" buried in executable code, but comments should be included referring to the source of the empirical formula. Hard-coded numbers should never be passed through argument lists.
+
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+\subsection{Declaration for variables and constants}
+
+\subsubsection{Rules :}
+Variables used as constants should be declared with attribute PARAMETER and used always without copying to local variables, inorder to prevent from using different values for the same constant or changing it accidentally.
+\begin{itemize}
+\item Usage of the DIMENSION statement or attribute is required in declaration statements
+\item The ``::'' notation is quite useful to show that this program unit declaration part is written in standard FORTRAN syntax, even if there are no attributes to clarify the declaration section. Always use the notation $<$blank$>$::$<$three blanks$>$ to improve readability.
+\item Declare the length of a character variable using the CHARACTER (len=xxx) syntax
+\footnote { The len specifier is important because it is possible to have several kinds for characters (e.g. Unicode using two bytes per character, or there might be a different kind for Japanese e.g. NEC). }
+
+\item For all global data (in contrast to module data, that is all data that can be access by other module) must be accompanied with a comment field on the same line.
+\footnote {This allows a easy research of where and how a variable is declared using the unix command: ``grep var *90 |grep !:''. }
+\\
+For example:
+\begin{verbatim}
+REAL(wp), DIMENSION(jpi,jpj,jpk) :: ua & !: i-horizontal velocity (m/s)
+\end{verbatim}
+\end{itemize}
+
+\subsubsection{Implicit None:}
+ All subroutines and functions will include an IMPLICIT NONE statement.
+Thus all variables must be explicitly typed. It also allows the compiler to detect typographical errors in variable names.
+For modules, one IMPLICIT NONE statement in the modules definition section is needed. This also removes the need to have IMPLICIT NONE statements in any routines that are CONTAIN'd in the module.
+Improper data initialisation is another common source of errors.
+\footnote{A variable could contain an initial value you did not expect. This can happen for several reasons, e.g. the variable has never been assigned a value, its value is outdated, memory has been allocated for a pointer but you have forgotten to initialise the variable pointed to.}
+To avoid problems, initialise variables as close as possible to where they are first used.
+
+\subsubsection{Attributes:}
+$PRIVATE / PUBLIC$ :
+All resources of a module are $PUBLIC$ by default.
+A reason to store multiple routines and their data in a single module is that the scope of the data defined in the module can be limited to the routines which are in the same module. This is accomplished with the $PRIVATE$ attribute.\\
+$INTENT$ :
+All dummy arguments of a routine must include the $INTENT$ clause in their declaration in order to improve control of variables in routine calls.
+
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+\subsection{Headers}
+Prologues are not used in NEMO for now, although it may become an interesting tool in combination with ProTeX auto documentation script in the future.
+Rules to code the headers and layout of a module or a routine are illustrated in the example module available with the code : {\it NEMO/OPA\_SRC/module\_example}
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+\subsection{Interface blocks}
+Explicit interface blocks are required between routines if optional or keyword arguments are to be used. They also allow the compiler to check that the type, shape and number of arguments specified in the CALL are the same as those specified in the subprogram itself. FORTRAN 95 compilers can automatically provide explicit interface blocks for routines contained in a module.
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+\subsection{I/O Error Conditions}
+I/O statements which need to check an error condition will use the $iostat=$ construct instead of the outmoded end= and err=. \\
+Note that a 0 value means success, a positive value means an error has occurred, and a negative value means the end of record or end of file was encountered.
+ %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+\subsection{PRINT - ASCII output files}
+Output listing and errors are directed to $numout$ logical unit =6 and produces a file called $ocean.output$ (use ln\_prt to have one output per process in MPP). Logical $lwp$ variable allows for less verbose outputs.
+To output an error from a routine, one can use the following template:
+\begin{verbatim}
+ IF( nstop /= 0 .AND. lwp ) THEN ! error print
+ WRITE(numout,cform_err)
+ WRITE(numout,*) nstop, ' error have been found'
+ ENDIF
+\end{verbatim}
+
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+\subsection{Precision}
+Parameterizations should not rely on vendor-supplied flags to supply a default floating point precision or integer size. The F95$ KIND$ feature should be used instead. In order to improve portability between 32 and 64 bit platforms, it is necessary to make use of kinds by using a specific module ($OPA\_SRC/par\_kind.F90$) declaring the "kind definitions" to obtain the required numerical precision and range as well as the size of INTEGER. It should be noted that numerical constants need to have a suffix of \_$kindvalue$ to have the according size. \\
+Thus $wp$ being the "working precision" as declared in $OPA\_SRC/par\_kind.F90$, declaring real array $zpc$ will take the form:
+\begin{verbatim}
+ REAL(wp), DIMENSION(jpi,jpj,jpk) :: zpc ! power consumption
+\end{verbatim}
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+\subsection{Structures}
+The TYPE structure allowing to declare some variables is more often used in NEMO, especially in the modules dealing with reading fields, or interfaces.For example
+\begin{verbatim}
+ ! Definition of a tracer as a structure
+ TYPE PTRACER
+ CHARACTER(len = 20) :: sname !: short name
+ CHARACTER(len = 80 ) :: lname !: long name
+ CHARACTER(len = 20 ) :: unit !: unit
+ LOGICAL :: lini !: read in a file or not
+ LOGICAL :: lsav !: ouput the tracer or not
+ END TYPE PTRACER
+
+ TYPE(PTRACER) , DIMENSION(jptra) :: tracer
+\end{verbatim}
+
+ Missing rule on structure name??
+ %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+\section{Packages coding rules}
+
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+\subsection{Bounds checking}
+NEMO is able to run when an array bounds checking option is enabled (provided the cpp key $key\_vectopt\_loop$ is not defined). \\
+Thus, constructs of the following form are disallowed:
+\begin{verbatim}
+REAL(wp) :: arr(1)
+\end{verbatim}
+where "arr" is an input argument into which the user wishes to index beyond 1. Use of the (*) construct in array dimensioning is forbidden also because it effectively disables array bounds checking.
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+\subsection{Communication}
+A package should refer only to its own modules and subprograms and to those intrinsic functions included in the Fortran standard.\\
+All communication with the package will be through the argument list or namelist input.
+\footnote { The point behind this rule is that packages should not have to know details of the surrounding model data structures, or the names of variables outside of the package. A notable exception to this rule is model resolution parameters. The reason for the exception is to allow compile-time array sizing inside the package. This is often important for efficiency.}
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+\subsection{Error conditions}
+When an error condition occurs inside a package, a message describing what went wrong will be printed (see PRINT - ASCII output files). The name of the routine in which the error occurred must be included. It is acceptable to terminate execution within a package, but the developer may instead wish to return an error flag through the argument list, see $stpctl.F90$.
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+\subsection{Memory management}
+
+The main action is to identify and declare which arrays are PUBLIC and
+which are PRIVATE.\\ As of version 3.3.1 of NEMO, the use of static
+arrays (size fixed at compile time) has been deprecated. All module
+arrays are now declared ALLOCATABLE and allocated in either the
+$<$module\_name$>$\_alloc() or $<$module\_name$>$\_init()
+routines. The success or otherwise of each ALLOCATE must be checked
+using the $Stat=$ optional argument.\\
+
+In addition to arrays contained within modules, many routines in NEMO
+require local, ``workspace'' arrays to hold the intermediate results
+of calculations. In previous versions of NEMO, these arrays were
+declared in such a way as to be automatically allocated on the stack
+when the routine was called. An example of an automatic array is:
+\begin{verbatim}
+SUBROUTINE sub(n)
+ REAL :: a(n)
+ ...
+END SUBROUTINE sub
+\end{verbatim}
+The downside of this approach is that the program will crash if it
+runs out of stack space and the reason for the crash might not be
+obvious to the user.
+
+Therefore, as of version 3.3.1, the use of automatic arrays is
+deprecated. Instead, a new module, ``wrk\_nemo,'' has been introduced
+which contains 1-,2-,3- and 4-dimensional workspace arrays for use in
+subroutines. These workspace arrays should be used in preference to
+declaring new, local (allocatable) arrays whenever possible. The only
+exceptions to this are when workspace arrays with lower bounds other
+than 1 and/or with extent(s) greater than those in the {\it wrk\_nemo}
+module are required.\\
+
+The 2D, 3D and 4D workspace arrays in {\it wrk\_nemo} have extents
+$jpi$, $jpj$, $jpk$ and $jpts$ ($x$, $y$, $z$ and tracers) in the first,
+second, third and fourth dimensions, respectively. The 1D arrays are
+allocated with extent MAX($jpi\times jpj, jpk\times jpj, jpi\times
+jpk$).\\
+
+The REAL (KIND=$wp$) workspace arrays in {\it wrk\_nemo} are named
+e.g. $wrk\_1d\_1$, $wrk\_4d\_2$ etc. and should be accessed by USE'ing
+the {\it wrk\_nemo} module. Since these arrays are available to any
+routine, some care must be taken that a given workspace array is not
+already being used somewhere up the call stack. To help with this,
+{\it wrk\_nemo} also contains some utility routines; {\it
+ wrk\_in\_use()} and {\it wrk\_not\_released()}. The former first
+checks that the requested arrays are not already in use and then sets
+internal flags to show that they are now in use. The {\it
+ wrk\_not\_released()} routine un-sets those internal flags. A
+subroutine using this functionality for two, 3D workspace arrays named
+$zwrk1$ and $zwrk2$ will look something like:
+\begin{verbatim}
+SUBROUTINE sub()
+ USE wrk_nemo, ONLY: wrk_in_use, wrk_not_released
+ USE wrk_nemo, ONLY: zwrk1 => wrk_3d_5, zwrk2 => wrk_3d_6
+ !
+ IF(wrk_in_use(3, 5,6)THEN
+ CALL ctl_stop('sub: requested workspace arrays unavailable.')
+ RETURN
+ END IF
+ ...
+ ...
+ IF(wrk_not_released(3, 5,6)THEN
+ CALL ctl_stop('sub: failed to release workspace arrays.')
+ END IF
+ !
+END SUBROUTINE sub
+\end{verbatim}
+The first argument to each of the utility routines is the
+dimensionality of the required workspace (1--4). Following this there
+must be one or more integers identifying which workspaces are to be
+used/released.
+Note that, in the interests of keeping the code as simple as possible,
+there is no use of POINTERs etc. in the {\it wrk\_nemo}
+module. Therefore it is the responsibility of the developer to ensure
+that the arguments to {\it wrk\_in\_use()} and {\it
+ wrk\_not\_released()} match the workspace arrays actually being used
+by the subroutine.\\
+
+If a workspace array is required that has extent(s) less than those of
+the arrays in the {\it wrk\_nemo} module then the advantages of
+implicit loops and bounds checking may be retained by defining a
+pointer to a sub-array as follows:
+\begin{verbatim}
+SUBROUTINE sub()
+ USE wrk_nemo, ONLY: wrk_in_use, wrk_not_released
+ USE wrk_nemo, ONLY: wrk_3d_5
+ !
+ REAL(wp), DIMENSION(:,:,:), POINTER :: zwrk1
+ !
+ IF(wrk_in_use(3, 5)THEN
+ CALL ctl_stop('sub: requested workspace arrays unavailable.')
+ RETURN
+ END IF
+ !
+ zwrk1 => wrk_3d_5(1:10,1:10,1:10)
+ ...
+END SUBROUTINE sub
+\end{verbatim}
+Here, instead of ``use associating'' the variable $zwrk1$ with the
+array $wrk\_3d\_5$ (as in the first example), it is explicitly
+declared as a pointer to a 3D array. It is then associated with a
+sub-array of $wrk\_3d\_5$ once the call to {\it wrk\_in\_use()} has
+completed successfully. Note that in F95 (to which NEMO conforms) it
+is not possible for either the upper or lower array bounds of the
+pointer object to differ from those of the target array.\\
+
+In addition to the REAL (KIND=$wp$) workspace arrays, {\it wrk\_nemo}
+also contains 2D integer arrays and 2D REAL arrays with extent ($jpi$,
+$jpk$), {\it i.e.} $xz$. The utility routines for the integer
+workspaces are {\it iwrk\_in\_use()} and {\it iwrk\_not\_released()}
+while those for the $xz$ workspaces are {\it wrk\_in\_use\_xz()}
+and {\it wrk\_not\_released\_xz()}.
+
+Should a call to one of the {\it wrk\_in\_use()} family of utilities
+fail, an error message is printed along with a table showing which of
+the workspace arrays are currently in use. This should enable the
+developer to choose alternatives for use in the subroutine being
+worked on.\\
+
+When compiling NEMO for production runs, the calls to {\it
+ wrk\_in\_use()}/{\it wrk\_not\_released()} can be reduced to stubs
+that just return $.$FALSE$.$ by setting the cpp key
+{\it key\_no\_workspace\_check}. These stubs may then be inlined (and
+thus effectively removed altogether) by setting appropriate compiler
+flags (e.g. ``-finline'' for the Intel compiler or ``-Q'' for the IBM
+compiler).
+
+ %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+\subsection{Optimisation}
+
+Considering the new computer architecture, optimisation cannot be considered independently from the computer type.
+In NEMO, portability is a priority, before any too specific optimisation.
+Some tools are available to help: \\
+For vector computers:
+\begin{itemize}
+\item using $key\_vectopt\_loop$ allows to unroll a loop
+\end{itemize}
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+\subsection{Package attribute: $PRIVATE, PUBLIC, USE, ONLY$}
+Module variables and routines should be encapsulated by using the PRIVATE attribute. What shall be used outside the module can be declared PUBLIC instead. Use USE with the ONLY attribute to specify which of the variables, type definitions etc. defined in a module are to be made available to the using routine.
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+\subsection {Parallelism: using MPI}
+NEMO is written in order to be able to run on one processor, or on one or more using MPI (i.e. activating the cpp key $key\_mpp\_mpi$. The domain decomposition divides the global domain in cubes (see NEMO reference manual). Whilst coding a new development, the MPI compatibility has to be taken in account (see $LBC/lib\_mpp.F90$) and should be tested. By default, the $x$-$z$ part of the decomposition is chosen to be as square as possible. However, this may be overriden by specifying the number of subdomains in latitude and longitude in the nammpp section of the namelist file.
+
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+\section{Features to be avoided}
+
+The code must follow the current standards of FORTRAN and ANSI C. In particular, the code should not produce any WARNING at compiling phase, so that users can be easily alerted of potential bugs when some appear in their new developments. ).
+Below is a list of features to avoid:
+\begin{itemize}
+\item COMMON blocks (use the declaration part of MODULEs instead)
+\item EQUIVALENCE (use POINTERs or derived data types instead to form data structures)
+\item Assigned and computed GOTOs (use the CASE construct instead)
+\item Arithmetic IF statements ( use the block IF, ELSE, ELSEIF, ENDIF or SELECT CASE construct instead)
+\item Labeled DO constructs (use unlabeled END DO instead)
+\item FORMAT statements (use character parameters or explicit format- specifiers inside the READ or WRITE statement instead)
+\item GOTO and CONTINUE statement (use IF, CASE, DO WHILE, EXIT or CYCLE statements or a contained
+\item PAUSE
+\item ENTRY statements: a subprogram must only have one entry point.
+\item RETURN Ð it is obsolete and so not necessary at the end of program units
+\item STATEMENT FUNCTION
+ \item Avoid functions with side effects.
+\footnote{ First, the code is easier to understand, if you can rely on the rule that functions don't change their arguments, second, some compilers generate more efficient code for PURE (in FORTRAN 95 there are the attributes PURE and ELEMENTAL) functions, because they can store the arguments in different places. This is especially important on massive parallel and as well on vector machines. }
+\item DATA and BLOCK DATA - (use initialisers)
+\end{itemize}
+
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+
+% \printindex
+% \input NEMO_coding.conv.ind
+
+\end{document}
Index: /branches/UKMO/dev_r5518_GO6_under_ice_relax_dr_hook/DOC/TexFiles/Biblio/Biblio.bib
===================================================================
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+++ /branches/UKMO/dev_r5518_GO6_under_ice_relax_dr_hook/DOC/TexFiles/Biblio/Biblio.bib (revision 11643)
@@ -0,0 +1,3032 @@
+This file was created with JabRef 2.2.
+Encoding: UTF8
+
+@STRING{AP = {Academic Press}}
+
+@STRING{AREPS = {Annual Review of Earth Planetary Science}}
+
+@STRING{ARFM = {Annual Review of Fluid Mechanics}}
+
+@STRING{ASL = {Atmospheric Science Letters}}
+
+@STRING{AW = {Addison-Wesley}}
+
+@STRING{CP = {Clarendon Press}}
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+
+@STRING{CUP = {Cambridge University Press}}
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+@STRING{CSR = {Cont. Shelf Res.}}
+
+@STRING{D = {Dover Publications}}
+
+@STRING{DAO = {Dyn. Atmos. Ocean}}
+
+@STRING{DSR = {Deep-Sea Res.}}
+
+@STRING{E = {Eyrolles}}
+
+@STRING{EFM ={Environ. Fluid Mech.}}
+
+@STRING{GRL = {Geophys. Res. Let.}}
+
+@STRING{I = {Interscience}}
+
+@STRING{JAOT = {J. Atmos. Ocean Tech.}}
+
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+@STRING{JSC = {J. Supercomputing.}}
+
+@STRING{JWS = {John Wiley and Sons}}
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+@STRING{MGH = {McGraw-Hill}}
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+@STRING{MWR = {Mon. Wea. Rev.}}
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+@STRING{Nature = {Nat.}}
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+@STRING{NH = {North-Holland}}
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+@STRING{OS = {Ocean Sc.}}
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+}
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+}
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+}
+
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+}
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+ url = {http://dx.doi.org/10.1029/94RG01872}
+}
+
+@ARTICLE{Latif_al_JC06,
+ author = {M. Latif and C. Böning and J. Willebrand and A. Biastoch and J. Dengg
+ and N. Keenlyside and U. Schweckendiek and G. Madec},
+ title = {Is the Thermohaline Circulation Changing?},
+ journal = JC,
+ year = {2006},
+ volume = {19},
+ pages = {4631--4637}
+}
+
+@PHDTHESIS{Lazar_PhD97,
+ author = {A. Lazar},
+ title = {La branche froide de la circulation thermohaline - sensibilit\'{e}
+ \`{a} la diffusion turbulente dans un mod\`{e}le de circulation g\'{e}n\'{e}rale
+ id\'{e}alis\'{e}e},
+ school = {Universit\'{e} Pierre et Marie Curie, Paris, France, 200pp},
+ year = {1997}
+}
+
+@ARTICLE{Lazar_al_JPO99,
+ author = {A. Lazar and G. Madec and P. Delecluse},
+ title = {The Deep Interior Downwelling, the Veronis Effect, and Mesoscale
+ Tracer Transport Parameterizations in an OGCM},
+ journal = JPO,
+ year = {1999},
+ volume = {29}, number = {11},
+ pages = {2945--2961},
+}
+
+@ARTICLE{Le_Sommer_al_OM09,
+ author = {J. {Le Sommer} and T. Penduff and S. Theetten and G. Madec and B.
+ Barnier},
+ title = {How momentum advection schemes influence current-topography interactions
+ at eddy permitting resolution},
+ journal = OM,
+ year = {2009},
+ volume = {29}, number = {1},
+ pages = {1--14},
+ doi = {10.1016/j.ocemod.2008.11.007},
+ url = {http://dx.doi.org/10.1016/j.ocemod.2008.11.007}
+}
+
+@PHDTHESIS{Leclair_PhD2010,
+ author = {M. Leclair},
+ title = {introduction d'une coordonn\'{e}e verticale arbitrairement Lagrangienne
+ Eul\'{e}rienne dans le code NEMO, 180pp.},
+ school = {Universit\'{e} Pierre and Marie Curie},
+ year = {2010}
+}
+
+@ARTICLE{Leclair_Madec_OM09,
+ author = {M. Leclair and G. Madec},
+ title = {A conservative leap-frog time stepping method},
+ journal = OM,
+ year = {2009},
+ volume = {30}, number = {2-3},
+ pages = {88-94},
+ doi = {10.1016/j.ocemod.2009.06.006},
+ url = {http://dx.doi.org/}
+}
+
+@ARTICLE{Leclair_Madec_OM10s,
+ author = {M. Leclair and G. Madec},
+ title = {$\tilde{z}$-coordinate, an Arbitrary Lagrangian-Eulerian coordinate separating high and low frequency},
+ journal = OM,
+ year = {2010},
+ pages = {submitted},
+}
+
+@ARTICLE{Lengaigne_al_JC03,
+ author = {M. Lengaigne and J.-P. Boulanger and C. Menkes and G. Madec and P.
+ Delecluse and E. Guilyardi, and J. Slingo},
+ title = {The March 1997 Westerly Wind Event and the onset of the 1997/98 El
+ Niño: Understanding the role of the atmospheric},
+ journal = JC,
+ year = {2003},
+ volume = {16}, number = {20},
+ pages = {3330--3343}
+}
+
+@ARTICLE{Lengaigne_al_JGR02,
+ author = {M. Lengaigne and J.-P. Boulanger and C. Menkes and S. Masson and
+ G. Madec and P. Delecluse},
+ title = {Ocean response to the March 1997 Westerly Wind Event},
+ journal = JGR,
+ year = {2002},
+ doi = {10.1029/2001JC000841},
+ url = {http://dx.doi.org/10.1029/2001JC000841}
+}
+
+@ARTICLE{Lengaigne_al_JGR03,
+ author = {M. Lengaigne and G. Madec and G. Alory and C. Menkes},
+ title = {Sensitivity of the tropical Pacific Ocean to isopycnal diffusion
+ on tracer and dynamics},
+ journal = JGR,
+ year = {2003},
+ volume = {108}, number = {C11},
+ pages = {3345},
+ doi = {10.1029/2002JC001704},
+ url = {http://dx.doi.org/10.1029/2002JC001704}
+}
+
+@ARTICLE{Lengaigne_al_GRL09,
+ author = {M. Lengaigne and G. Madec and L. Bopp and C. Menkes and O. Aumont and P. Cadule},
+ title = {Bio-physical feedbacks in the Arctic Ocean using an Earth System model},
+ journal = GRL,
+ year = {2009},
+ volume = {36},
+ pages = {L21602},
+ doi = {10.1029/2009GL040145},
+ url = {http://dx.doi.org/10.1029/2009GL040145}
+}
+
+@ARTICLE{Lengaigne_al_CD07,
+ author = {M. Lengaigne and C. Menkes and O. Aumont and T. Gorgues and L. Bopp and J.-M. Andr\'{e} G. Madec},
+ title = {Bio-physical feedbacks on the tropical Pacific climate in a Coupled
+ General Circulation Model},
+ journal = CD,
+ year = {2007},
+ volume = {28},
+ pages = {503--516}
+}
+
+@ARTICLE{Leonard1991,
+ author = {B. P. Leonard},
+ title = {The ULTIMATE conservative difference scheme applied to unsteady one--dimensional advection},
+ journal = {Computer Methods in Applied Mechanics and Engineering},
+ year = {1991},
+ pages = {17--74}
+}
+
+@TECHREPORT{Leonard_Rep88,
+ author = {B. P. Leonard},
+ title = {Universal limiter for transient interpolation modelling of the advective transport equations},
+ institution = {Technical Memorandum TM-100916 ICOMP-88-11, NASA},
+ year = {1988}
+}
+
+@ARTICLE{Leonard1979,
+ author = {B. P. Leonard},
+ title = {A stable and accurate convective modelling procedure based on quadratic
+ upstream interpolation},
+ journal = {Computer Methods in Applied Mechanics and Engineering},
+ year = {1979},
+ volume = {19},
+ pages = {59--98},
+}
+
+@TECHREPORT{Levier2007,
+ author = {B. Levier and A.-M. Tr\'{e}guier and G. Madec and V. Garnier},
+ title = {Free surface and variable volume in the NEMO code},
+ institution = {MERSEA MERSEA IP report WP09-CNRS-STR-03-1A, 47pp, available on the
+ NEMO web site},
+ year = {2007}
+}
+
+@BOOK{levitus82,
+ title = {Climatological Atlas of the world ocean},
+ publisher = {NOAA professional paper No. 13, 174pp},
+ year = {1982},
+ author = {S Levitus },
+ pages = {173 pp}
+}
+
+@ARTICLE{Li_Garrett_JMR93,
+ author = {M. Li and C. Garrett},
+ title = {Cell merging and the jet/downwelling ratio in Langmuir circulation},
+ journal = JMR,
+ year = {1993},
+ volume = {51},
+ pages = {737--769}
+}
+
+@TECHREPORT{Lott1989,
+ author = {F. Lott and G. Madec},
+ title = {Implementation of bottom topography in the Ocean General Circulation
+ Model OPA of the LODYC: formalism and experiments.},
+ institution = {LODYC, France, 36pp.},
+ year = {1989},
+ number = {3}
+}
+
+@ARTICLE{Lott_al_OM90,
+ author = {F. Lott and G. Madec and J. Verron},
+ title = {Topographic experiments in an Ocean General Circulation Model},
+ journal = OM,
+ year = {1990},
+ volume = {88},
+ pages = {1--4}
+}
+
+@ARTICLE{Luo_al_JC05,
+ author = {J.-J. Luo and S. Masson and E. Roeckner and G. Madec and T. Yamagata},
+ title = {Reducing climatology bias in an ocean-atmosphere CGCM with improved
+ coupling physics},
+ journal = JC,
+ year = {2005},
+ volume = {18}, number = {13},
+ pages = {2344--2360}
+}
+
+@BOOK{Madec_Bk08,
+ title = {NEMO ocean engine},
+ publisher = {Note du P\^ole de mod\'{e}lisation, Institut Pierre-Simon Laplace
+ (IPSL), France, No 27, ISSN No 1288-1619},
+ year = {2008},
+ author = {G. Madec}
+}
+
+@BOOK{Madec_HDR01,
+ title = {Le Cycle des Masses d'Eau Oc\'{e}aniqueset sa variabilit\'{e} dans le Syst\'{e}me Climatique},
+ year = {2001},
+ author = {G. Madec},
+ pages = {63pp.},
+ series = {Habilitation \'{a} Diriger des Recherches, Universit\'{e} Pierre et Marie Curie}
+}
+
+@PHDTHESIS{Madec_PhD90,
+ author = {G. Madec},
+ title = {La formation d'eau profonde et son impact sur la circulation r\'{e}gionale
+ en M\'{e}diterran\'{e}e Occidentale - une approche num\'{e}rique},
+ school = {Universit\'{e} Pierre et Marie Curie, Paris, France, 194pp.},
+ year = {1990},
+ month = {2 mai}
+}
+
+@ARTICLE{Madec_al_DAO91,
+ author = {G. Madec and M. Chartier and M. Cr\'{e}pon},
+ title = {Effect of thermohaline forcing variability on deep water formation
+ in the Northwestern Mediterranean Sea - a high resulution three-dimensional
+ study},
+ journal = DAO,
+ year = {1991},
+ volume = {15},
+ pages = {301--332}
+}
+
+@ARTICLE{Madec_al_JPO91,
+ author = {G. Madec and M. Chartier and P. Delecluse and M. Cr\'{e}pon},
+ title = {A three-dimensional numerical study of deep water formation in the Northwestern Mediterranean Sea .},
+ journal = JPO,
+ year = {1991},
+ volume = {21},
+ pages = {1349--1371}
+}
+
+@INBOOK{Madec_Crepon_Bk91,
+ chapter = {Thermohaline-driven deep water formation in the Northwestern Mediterranean
+ Sea},
+ pages = {241--265},
+ title = {Deep convection and deep water formation in the oceans},
+ publisher = {Elsevier Oceanographic Series, P.C. Chu and J.C. Gascard (Eds.)},
+ year = {1991},
+ author = {G. Madec and M. Cr\'{e}pon}
+ }
+
+@ARTICLE{Madec1997,
+ author = {G. Madec and P. Delecluse},
+ title = {The OPA/ARPEGE and OPA/LMD Global Ocean-Atmosphere Coupled Model},
+ journal = {Int. WOCE Newsletter},
+ year = {1997},
+ volume = {26},
+ pages = {12--15}
+}
+
+@TECHREPORT{Madec1998,
+ author = {G. Madec and P. Delecluse and M. Imbard and C. Levy},
+ title = {OPA 8 Ocean General Circulation Model - Reference Manual},
+ institution = {LODYC/IPSL Note 11},
+ year = {1998}
+}
+
+@ARTICLE{Madec_Imbard_CD96,
+ author = {G Madec and M Imbard},
+ title = {A global ocean mesh to overcome the north pole singularity},
+ journal = CD,
+ year = {1996},
+ volume = {12},
+ pages = {381--388}
+}
+
+@ARTICLE{Madec_al_JPO96,
+ author = {G. Madec and F. Lott and P. Delecluse and M. Cr\'{e}pon},
+ title = {Large-Scale Preconditioning of Deep-Water Formation in the Northwestern
+ Mediterranean Sea},
+ journal = JPO,
+ year = {1996},
+ volume = {26}, number = {8},
+ pages = {1393--1408},
+}
+
+@ARTICLE{Madec_al_OM88,
+ author = {G. Madec and C. Rahier and M. Chartier},
+ title = {A comparison of two-dimensional elliptic solvers for the streamfunction
+ in a multilevel OGCM},
+ journal = OM,
+ year = {1988},
+ volume = {78},
+ pages = {1-6}
+}
+
+@ARTICLE{Maes_al_CD98,
+ author = {C. Maes and P. Delecluse and G. Madec},
+ title = {Impact of westerly wind bursts on the warm pool of the TOGA-COARE
+ domain in an OGCM},
+ journal = CD,
+ year = {1998},
+ volume = {14},
+ pages = {55--70}
+}
+
+@ARTICLE{Maes_al_MWR97,
+ author = {C. Maes and G. Madec and P. Delecluse},
+ title = {Sensitivity of an Equatorial Pacific OGCM to the lateral diffusion},
+ journal = MWR,
+ year = {1997},
+ volume = {125}, number = {5},
+ pages = {958--971}
+}
+
+@ARTICLE{Maggiore_al_PCE98,
+ author = {A. Maggiore and M. Zavatarelli and M. G. Angelucci and N. Pinardi},
+ title = {Surface heat and water fluxes in the Adriatic Sea: seasonal and interannual variability},
+ journal = {Phys Chem Earth},
+ year = {1998},
+ volume = {23},
+ pages = {561--567}
+}
+
+@ARTICLE{Maltrud1998,
+ author = {M. E. Maltrud and R. D. Smith and A. J. Semtner and R. C. Malone},
+ title = {Global eddy-resolving ocean simulations driven by 1985-1995 atmospheric
+ winds},
+ journal = JGR,
+ year = {1998},
+ volume = {103}, number = {C13},
+ pages = {30,825--30,854}
+}
+
+@ARTICLE{Marchesiello2001,
+ author = { P. Marchesiello and J. Mc Williams and A. Shchepetkin },
+ title = {Open boundary conditions for long-term integrations of Regional Oceanic
+ Models},
+ journal = OM,
+ year = {2001},
+ volume = {3},
+ pages = {1--20}
+}
+
+
+@article{Martin_Adcroft_OM10,
+author = {T. Martin and A. Adcroft},
+title = {Parameterizing the fresh-water flux from land ice to ocean with interactive icebergs in a coupled climate model},
+journal = OM,
+year = {2010},
+volume = {34}, number = {3--4},
+pages = {111--124},
+issn = {1463-5003},
+doi = {10.1016/j.ocemod.2010.05.001},
+url = {http://dx.doi.org/10.1016/j.ocemod.2010.05.001}
+}
+
+@ARTICLE{Marsaleix_al_OM08,
+ author = {P. Marsaleix and F. Auclair and J. W. Floor and M. J. Herrmann and
+ C. Estournel and I. Pairaud and C. Ulses},
+ title = {Energy conservation issues in sigma-coordinate free-surface ocean
+ models},
+ journal = OM,
+ year = {2008},
+ volume = {20}, number = {1},
+ pages = {61--89},
+ doi = {10.1016/j.ocemod.2007.07.005},
+ url = {http://dx.doi.org/10.1016/j.ocemod.2007.07.005}
+}
+
+@BOOK{MIT-GCM_2004,
+ title = {MIT-gcm User Manual},
+ year = {2004},
+ editor = {MIT Department of EAPS},
+ author = {J. Marshall and A. Adcroft and J.-M. Campin and P. Heimbach and A.
+ Molod and S. Dutkiewicz and H. Hill and M. Losch and B. Fox-Kemper
+ and D. Menemenlis and D. Ferreira and E. Hill and M. Follows and
+ C. Hill and C. Evangelinos and G. Forget}
+ }
+
+@PHDTHESIS{Marti_PhD92,
+ author = {O. Marti},
+ title = {Etude de l'oc\'{e}an mondial : mod\'{e}lisation de la circulation
+ et du transport de traceurs anthropog\'{e}niques},
+ school = {Universit\'{e} Pierre et Marie Curie, Paris, France, 201pp},
+ year = {1992}
+}
+
+@ARTICLE{Marti_al_CD10,
+ author = {O. Marti and P. Braconnot and J.-L. Dufresne and J. Bellier and R.
+ Benshila and S. Bony and P. Brockmann and P. Cadule and A. Caubel
+ and F Codron and S. Denvil and L. Fairhead and T. Fichefet and M.-A.
+ Filiberti and M.-A. Foujols and P. Friedlingstein and H. Goosse and
+ J.-Y. Grandpeix and E. Guilyardi and F. Hourdin and G. Krinner and
+ C. L\'{e}vy and G. Madec and J. Mignot and I. Musat and D. Swingedouw
+ and C. Talandier},
+ title = {Key features of the IPSL ocean atmosphere model and its sensitivity
+ to atmospheric resolution},
+ journal = CD,
+ year = {2010},
+ volume = {34}, number = {1},
+ pages = {1--26},
+ doi = {10.1007/s00382-009-0640-6},
+ url = {http://dx.doi.org/10.1007/s00382-009-0640-6}
+}
+
+@ARTICLE{Marti_al_JGR92,
+ author = {O. Marti and G. Madec and P. Delecluse},
+ title = {Comment on "Net diffusivity in ocean general circulation models with
+ nonuniform grids" by F. L. Yin and I. Y. Fung},
+ journal = JGR,
+ year = {1992},
+ volume = {97},
+ pages = {12,763--12,766}
+}
+
+@INBOOK{Masson_al_Bk08,
+ chapter = {OPA9 - French experiments on the Earth Simulator and Teraflop Workbench
+ tunings},
+ pages = {25-34},
+ title = {In High Performance computing on Vector System 2007, Stuttgart, Germany},
+ publisher = {Springer-Verlag},
+ year = {2008},
+ editor = {Resch M, Roller S, Lammers P, Furui T, Galle M, Bez W},
+ author = {S. Masson and M.-A. Foujols and P. Klein and G. Madec and L. Hua and M. Levy
+ and H. Sasaki and K. Takahashi and F. Svensson},
+ doi = {10.1007/978-3-540-74384-2},
+ url = {http://dx.doi.org/10.1007/978-3-540-74384-2}
+}
+
+@ARTICLE{Masson_al_GRL05,
+ author = {S. Masson and J.-J. Luo and G. Madec and J. Vialard and F. Durand
+ and S. Gualdi and E. Guilyardi and S. Behera and P. Delecluse and
+ A. Navarra and T. Yamagata},
+ title = {Impact of barrier layer on winter-spring variability of the South-Eastern
+ Arabian Sea},
+ journal = GRL,
+ year = {2005},
+ volume = {32},
+ pages = {L07703},
+ doi = {10.1029/2004GL021980},
+ url = {http://dx.doi.org/10.1029/2004GL021980}
+}
+
+@ARTICLE{McDougall1987,
+ author = {T. J. McDougall},
+ title = {Neutral Surfaces},
+ journal = JPO,
+ year = {1987},
+ volume = {17}, number = {11},
+ pages = {1950--1964},
+}
+
+@ARTICLE{McDougall_Taylor_JMR84,
+ author = {T. J. McDougall and J. R. Taylor},
+ title = {Flux measurements across a finger interface at low values of the stability ratio},
+ journal = JMR,
+ year = {1984},
+ volume = {42},
+ pages = {1--14}
+}
+
+@ARTICLE{Mellor_Blumberg_JPO04,
+author = {G. Mellor and A. Blumberg},
+title = {Wave Breaking and Ocean Surface Layer Thermal Response},
+journal = JPO,
+volume = {34}, number = {3},
+pages = {693--698},
+year = {2004},
+doi = {10.1175/2517.1},
+URL = {http://journals.ametsoc.org/doi/abs/10.1175/2517.1}
+}
+
+@ARTICLE{Mellor_Yamada_1982,
+ author = {G. L. Mellor and T. Yamada},
+ title = {Development of a turbulence closure model for geophysical fluid problems},
+ journal = RGSP,
+ year = {1982},
+ volume = {20},
+ pages = {851--875}
+}
+
+@ARTICLE{Menkes_al_JPO06,
+ author = {C. Menkes and J. Vialard and S C. Kennan and J.-P. Boulanger and G. Madec},
+ title = {A modelling study of the three-dimensional heat budget of Tropical
+ Instability Waves in the Equatorial Pacific},
+ journal = JPO,
+ year = {2006},
+ volume = {36}, number = {5},
+ pages = {847--865}
+}
+
+@ARTICLE{Merryfield1999,
+ author = {W. J. Merryfield and G. Holloway and A. E. Gargett},
+ title = {A Global Ocean Model with Double-Diffusive Mixing},
+ journal = JPO,
+ year = {1999},
+ volume = {29}, number = {6},
+ pages = {1124--1142}
+}
+
+@BOOK{Mesinger_Arakawa_Bk76,
+ title = {Numerical methods used in Atmospheric models},
+ publisher = {GARP Publication Series No 17},
+ year = {1976},
+ author = {F. Mesinger and A. Arakawa}
+}
+
+@ARTICLE{Morel_JGR88,
+ author = {A. Morel},
+ title = {Optical modeling of the upper ocean in relation to its biogenous matter content (Case I waters)},
+ journal = JGR,
+ year = {1988},
+ volume = {93},
+ pages = {10,749--10,768}
+}
+
+@ARTICLE{Morel_Maritorena_JGR01,
+ author = {A. Morel and S. Maritorena},
+ title = {Bio-optical properties of oceanic waters: a reappraisal},
+ journal = JGR,
+ year = {2001},
+ volume = {106}, number = {C4},
+ pages = {7163--7180}
+}
+
+@ARTICLE{Moun_al_JPO02,
+ author = {J.N. Moum and D.R. Caldwell and J.D. Nash and G.D. Gunderson},
+ title = {Observations of boundary mixing over the continental slope},
+ journal = JPO,
+ year = {2002},
+ volume = {32}, number = {7},
+ pages = {2113--2130}
+}
+
+@ARTICLE{Murray_JCP96,
+ author = {R. J. Murray},
+ title = {Explicit Generation of Orthogonal Grids for Ocean Models},
+ journal = JCP,
+ year = {1996},
+ volume = {126}, number = {2},
+ pages = {251--273},
+}
+
+@ARTICLE{Oddo_al_OS09,
+ author = {P. Oddo and M. Adani and N. Pinardi and C. Fratianni and M. Tonani and D. Pettenuzzo},
+ title = {A nested Atlantic-Mediterranean Sea general circulation model for operational forecasting},
+ journal = OS,
+ year = {2009},
+ volume = {5},
+ pages = {1--13},
+}
+
+@PHDTHESIS{Olivier_PhD01,
+ author = {F. Olivier},
+ title = {Etude de l'activit\'{e} biologique et de la circulation oc\'{e}anique
+ dans un jet g\'{e}ostrophique: le front Alm\'{e}ria-Oran},
+ school = {Universit\'{e} Pierre et Marie Curie, Paris, France},
+ year = {2001}
+}
+
+@ARTICLE{Osborn_JPO80,
+ author = {T.R. Osborn},
+ title = {Estimates of the local rate of vertical diﬀusion from dissipation measurements},
+ journal = JPO,
+ volume = {10},
+ pages = {83--89}
+}
+
+@ARTICLE{Pacanowski_Philander_JPO81,
+ author = {R.C. Pacanowski and S.G.H. Philander},
+ title = {Parameterization of Vertical Mixing in Numerical Models of Tropical Oceans},
+ journal = JPO,
+ year = {1981},
+ volume = {11}, number = {11},
+ pages = {1443--1451}
+}
+
+@ARTICLE{Pacanowski_Gnanadesikan_MWR98,
+ author = {R. C. Pacanowski and A. Gnanadesikan},
+ title = {Transient response in a z-level ocean model that resolves topography with partial-cells},
+ journal = MWR,
+ year = {1998},
+ volume = {126},
+ pages = {3248--3270}
+}
+
+@ARTICLE{Park_al_JC09,
+ author = {W. Park and N. Keenlyside and M. Latif and A. Str\¨{o}h and R. Redler and E. Roeckner and G. Madec},
+ title = {Tropical Pacific Climate and its Response to Global Warming in the
+ Kiel Climate Model},
+ journal = JC,
+ year = {2009},
+ volume = {22}, number = {1},
+ pages = {71--92},
+ doi = {10.1175/2008JCLI2261.1},
+ url = {http://dx.doi.org/10.1175/2008JCLI2261.1}
+}
+
+@ARTICLE{Paulson1977,
+ author = {C. A. Paulson and J. J. Simpson},
+ title = {Irradiance Measurements in the Upper Ocean},
+ journal = JPO,
+ year = {1977},
+ volume = {7}, number = {6},
+ pages = {952--956}
+}
+
+@ARTICLE{Payne_JAS72,
+ author = {R. E. Payne},
+ title = {Albedo of the Sea Surface},
+ journal = JAS,
+ year = {1972},
+ volume = {29},
+ pages = {959--970}
+}
+
+@ARTICLE{Penduff_al_OM06,
+ author = {T. Penduff and B. Barnier and J.-M. Molines and G. Madec},
+ title = {On the use of current meter data to assess the realism of ocean model simulations},
+ journal = OM,
+ year = {2006},
+ volume = {11}, number = {3--4},
+ pages = {399--416}
+}
+
+@ARTICLE{Penduff_al_JGR00,
+ author = {T. Penduff and B. Barnier and A. Colin de Verdi\`{e}re},
+ title = {Self-adapting open boundaries for a regional model of the eastern North Atlantic},
+ journal = JGR,
+ year = {2000},
+ volume = {105},
+ pages = {11,279--11,297}
+}
+
+@ARTICLE{Penduff_al_OS07,
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+
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+
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+ pages = {L14314},
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+ url = {http://dx.doi.org/10.1029/2004GL019764}
+}
+
+@ARTICLE{Rodgers_al_GRL03,
+ author = {K. Rodgers and B. Blanke and G. Madec and O. Aumont and P. Ciais and J.-C. Dutay},
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+ volume = {30}, number = {2},
+ doi = {10.1029/2002GL016003},
+ url = {http://dx.doi.org/10.1029/2002GL016003}
+}
+
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+ author = {W. Rodi},
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+ journal = JGR,
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+}
+
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+
+@ARTICLE{Roullet_Madec_JGR00,
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+ for ocean general circulation models},
+ journal = JGR,
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+}
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+ journal = JAS,
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+}
+
+@ARTICLE{Sarmiento1982,
+ author = {J. L. Sarmiento and K. Bryan},
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+ journal = JGR,
+ year = {1982},
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+
+@BOOK{Shchepetkin_McWilliams_Bk08,
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+ pages = {784}
+}
+
+@ARTICLE{Shchepetkin_McWilliams_OM05,
+ author = {A. F. Shchepetkin and J. C. McWilliams},
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+
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+}
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+ year = {1992}
+}
+
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+}
+
+@ARTICLE{Speich_al_JPO96,
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+}
+
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+}
+
+@ARTICLE{St_Laurent_Nash_DSR04,
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+}
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+}
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+}
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+}
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+}
+
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+ author = {R. Timmermann and H. Goosse and G. Madec and T. Fichefet and C. \'{E}the and V. Duli\`{e}re},
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+}
+
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+}
+
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+}
+
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+}
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+ author = {A.-M. Tr\'{e}guier and M. H. England and S. R. Rintoul and G. Madec and J. Le Sommer and J.-M. Molines},
+ title = {Southern Ocean overturning across streamlines in an eddying simulation of the Antarctic Circumpolar Current},
+ journal = OS,
+ year = {2007},
+ volume = {4},
+ pages = {653--698}
+}
+
+@ARTICLE{Treguier_al_OD06,
+ author = {A.-M. Tr\'{e}guier and C. Gourcuff and P. Lherminier and H. Mercier and B. Barnier
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+ volume = {56},
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+ url = {http://dx.doi.org/10.1007/s10236-006-0069-y}
+}
+
+@ARTICLE{Treguier1997,
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+ year = {1997},
+ volume = {27}, number = {4},
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+}
+
+@ARTICLE{Umlauf_Burchard_JMS03,
+ author = {L. Umlauf and H. Burchard},
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+}
+
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+ author = {L. Umlauf and H. Burchard},
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+ year = {2005},
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+}
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+}
+
+@TECHREPORT{Valcke_al_Rep00,
+ author = {S. Valcke and L. Terray and A. Piacentini },
+ title = {The OASIS Coupled User Guide Version 2.4},
+ institution = {CERFACS},
+ year = {2000},
+ number = {TR/CMGC/00-10}
+}
+
+@ARTICLE{Vancoppenolle_al_OM09b,
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+}
+
+@ARTICLE{Vancoppenolle_al_OM09a,
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+ G. Madec and M. A. Morales Maqueda},
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+ sea ice. 1. Model description and validation},
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+ year = {2009},
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+ doi = {10.1016/j.ocemod.2008.10.005},
+ url = {http://dx.doi.org/10.1016/j.ocemod.2008.10.005}
+}
+
+@ARTICLE{Vialard_al_JPO01,
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+}
+
+@ARTICLE{Warner_al_OM05,
+ author = {J. C. Warner and C. R. Sherwood and H. G. Arango and R. P. Signell},
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+ doi = {10.1016/j.ocemod.2003.12.003},
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+}
+
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+}
+
+@ARTICLE{Weaver_Eby_JPO97,
+ author = {A. J. Weaver and M. Eby},
+ title = {On the numerical implementation of advection schemes for use in conjuction
+ with various mixing parameterizations in the GFDL ocean model},
+ journal = JPO,
+ year = {1997},
+ volume = {27}
+}
+
+@ARTICLE{Webb_al_JAOT98,
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+ pages = {1171--1187},
+}
+
+@ARTICLE{White_al_JCP09,
+ author = {L. White and A. Adcroft and R. Hallberg},
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+ and generalized coordinates in ocean models},
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+ year = {2009},
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+ pages = {8665--8692}
+}
+
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+ author = {D. C. Wilcox},
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+}
+
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+ year = {2001},
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+}
+
+@ARTICLE{Williams_al_DAO09,
+ author = {P.D. Williams and E. Guilyardi and G. Madec and S. Gualdi and E. Scoccimarro},
+ title = {The role of mean ocean salinity on climate},
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+ year = {2010},
+ volume = {49}, number = {2-3},
+ pages = {108--123},
+ doi = {10.1016/j.dynatmoce.2009.02.001},
+ url = {http://dx.doi.org/10.1016/j.dynatmoce.2009.02.001}
+}
+
+@ARTICLE{Williams_al_GRL07,
+ author = {P.D. Williams and E. Guilyardi and R. Sutton and J.M. Gregory and G. Madec},
+ title = {A new feedback on climate change from the hydrological cycle},
+ journal = GRL,
+ year = {2007},
+ volume = {34},
+ pages = {L08706},
+ doi = {10.1029/2007GL029275},
+ url = {http://dx.doi.org/10.1029/2007GL029275}
+}
+
+@ARTICLE{Williams_al_CD06,
+ author = {P.D. Williams and E. Guilyardi and R. Sutton and J.M. Gregory and G. Madec},
+ title = {On the climate response of the low-latitude Pacific ocean to changes in the global freshwater cycle},
+ journal = CD,
+ year = {2006},
+ volume = {27},
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+}
+
+@ARTICLE{Zalesak_JCP79,
+ author = {S. T. Zalesak},
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+ journal = JCP,
+ year = {1979},
+ volume = {31}
+}
+
+@ARTICLE{Zhang_Endoh_JGR92,
+ author = {Zhang, R.-H. and Endoh, M.},
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+}
+
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+1 ExplicitGroup:Mechanisms\;2\;;
+2 ExplicitGroup:CRF\;0\;;
+2 ExplicitGroup:Water vapor\;0\;;
+2 ExplicitGroup:Atmos mechanisms\;0\;;
+1 ExplicitGroup:GCMs\;2\;;
+2 ExplicitGroup:Uncertainty\;0\;;
+2 ExplicitGroup:Momentum balance\;0\;;
+1 ExplicitGroup:Climate change\;0\;;
+2 ExplicitGroup:IPCC AR4\;0\;;
+1 ExplicitGroup:Analysis tools\;0\;;
+1 KeywordGroup:EG publis\;0\;author\;guilyardi\;0\;0\;;
+}
+
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+
+% ================================================================
+% Abstract (English / French)
+% ================================================================
+
+\chapter*{Abstract / R\'{e}sum\'{e}}
+
+\vspace{-40pt}
+
+{\small
+The ocean engine of NEMO (Nucleus for European Modelling of the Ocean) is a primitive
+equation model adapted to regional and global ocean circulation problems. It is intended to
+be a flexible tool for studying the ocean and its interactions with the others components of
+the earth climate system over a wide range of space and time scales.
+Prognostic variables are the three-dimensional velocity field, a linear
+or non-linear sea surface height, the temperature and the salinity. In the horizontal direction,
+the model uses a curvilinear orthogonal grid and in the vertical direction, a full or partial step
+$z$-coordinate, or $s$-coordinate, or a mixture of the two. The distribution of variables is a
+three-dimensional Arakawa C-type grid. Various physical choices are available to describe
+ocean physics, including TKE, GLS and KPP vertical physics. Within NEMO, the ocean is
+interfaced with a sea-ice model (LIM v2 and v3), passive tracer and biogeochemical models (TOP)
+and, via the OASIS coupler, with several atmospheric general circulation models. It also
+support two-way grid embedding via the AGRIF software.
+
+% ================================================================
+ \vspace{0.5cm}
+
+Le moteur oc\'{e}anique de NEMO (Nucleus for European Modelling of the Ocean) est un
+mod\`{e}le aux \'{e}quations primitives de la circulation oc\'{e}anique r\'{e}gionale et globale.
+Il se veut un outil flexible pour \'{e}tudier sur un vaste spectre spatiotemporel l'oc\'{e}an et ses
+interactions avec les autres composantes du syst\`{e}me climatique terrestre.
+Les variables pronostiques sont le champ tridimensionnel de vitesse, une hauteur de la mer
+lin\'{e}aire ou non, la temperature et la salinit\'{e}.
+La distribution des variables se fait sur une grille C d'Arakawa tridimensionnelle utilisant une
+coordonn\'{e}e verticale $z$ \`{a} niveaux entiers ou partiels, ou une coordonn\'{e}e s, ou encore
+une combinaison des deux. Diff\'{e}rents choix sont propos\'{e}s pour d\'{e}crire la physique
+oc\'{e}anique, incluant notamment des physiques verticales TKE, GLS et KPP. A travers l'infrastructure
+NEMO, l'oc\'{e}an est interfac\'{e} avec des mod\`{e}les de glace de mer, de biog\'{e}ochimie
+et de traceurs passifs, et, via le coupleur OASIS, \`{a} plusieurs mod\`{e}les de circulation
+g\'{e}n\'{e}rale atmosph\'{e}rique. Il supporte \'{e}galement l'embo\^{i}tement interactif de
+maillages via le logiciel AGRIF.
+}
+
+% ================================================================
+% Disclaimer
+% ================================================================
+\chapter*{Disclaimer}
+
+Like all components of NEMO, the ocean component is developed under the CECILL license,
+which is a French adaptation of the GNU GPL (General Public License). Anyone may use it
+freely for research purposes, and is encouraged to communicate back to the NEMO team
+its own developments and improvements. The model and the present document have been
+made available as a service to the community. We cannot certify that the code and its manual
+are free of errors. Bugs are inevitable and some have undoubtedly survived the testing phase.
+Users are encouraged to bring them to our attention. The author assumes no responsibility
+for problems, errors, or incorrect usage of NEMO.
+
+ \vspace{1cm}
+NEMO reference in papers and other publications is as follows:
+ \vspace{0.5cm}
+
+Madec, G., and the NEMO team, 2008: NEMO ocean engine.
+\textit{Note du P\^ole de mod\'{e}lisation}, Institut Pierre-Simon Laplace (IPSL), France,
+No 27, ISSN No 1288-1619.\\
+
+
+ \vspace{0.5cm}
+Additional information can be found on \href{http://www.nemo-ocean.eu/}{nemo-ocean.eu} website.
+ \vspace{0.5cm}
+
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+
+% ================================================================
+% Chapter Ñ Appendix A : Curvilinear s-Coordinate Equations
+% ================================================================
+\chapter{Curvilinear $s-$Coordinate Equations}
+\label{Apdx_A}
+\minitoc
+
+\newpage
+$\ $\newline % force a new ligne
+
+% ================================================================
+% Chain rule
+% ================================================================
+\section{The chain rule for $s-$coordinates}
+\label{Apdx_A_continuity}
+
+In order to establish the set of Primitive Equation in curvilinear $s$-coordinates
+($i.e.$ an orthogonal curvilinear coordinate in the horizontal and an Arbitrary Lagrangian
+Eulerian (ALE) coordinate in the vertical), we start from the set of equations established
+in \S\ref{PE_zco_Eq} for the special case $k = z$ and thus $e_3 = 1$, and we introduce
+an arbitrary vertical coordinate $a = a(i,j,z,t)$. Let us define a new vertical scale factor by
+$e_3 = \partial z / \partial s$ (which now depends on $(i,j,z,t)$) and the horizontal
+slope of $s-$surfaces by :
+\begin{equation} \label{Apdx_A_s_slope}
+\sigma _1 =\frac{1}{e_1 }\;\left. {\frac{\partial z}{\partial i}} \right|_s
+\quad \text{and} \quad
+\sigma _2 =\frac{1}{e_2 }\;\left. {\frac{\partial z}{\partial j}} \right|_s
+\end{equation}
+
+The chain rule to establish the model equations in the curvilinear $s-$coordinate
+system is:
+\begin{equation} \label{Apdx_A_s_chain_rule}
+\begin{aligned}
+&\left. {\frac{\partial \bullet }{\partial t}} \right|_z =
+\left. {\frac{\partial \bullet }{\partial t}} \right|_s
+ -\frac{\partial \bullet }{\partial s}\;\frac{\partial s}{\partial t} \\
+&\left. {\frac{\partial \bullet }{\partial i}} \right|_z =
+ \left. {\frac{\partial \bullet }{\partial i}} \right|_s
+ -\frac{\partial \bullet }{\partial s}\;\frac{\partial s}{\partial i}=
+ \left. {\frac{\partial \bullet }{\partial i}} \right|_s
+ -\frac{e_1 }{e_3 }\sigma _1 \frac{\partial \bullet }{\partial s} \\
+&\left. {\frac{\partial \bullet }{\partial j}} \right|_z =
+\left. {\frac{\partial \bullet }{\partial j}} \right|_s
+ - \frac{\partial \bullet }{\partial s}\;\frac{\partial s}{\partial j}=
+\left. {\frac{\partial \bullet }{\partial j}} \right|_s
+ - \frac{e_2 }{e_3 }\sigma _2 \frac{\partial \bullet }{\partial s} \\
+&\;\frac{\partial \bullet }{\partial z} \;\; = \frac{1}{e_3 }\frac{\partial \bullet }{\partial s} \\
+\end{aligned}
+\end{equation}
+
+In particular applying the time derivative chain rule to $z$ provides the expression
+for $w_s$, the vertical velocity of the $s-$surfaces referenced to a fix z-coordinate:
+\begin{equation} \label{Apdx_A_w_in_s}
+w_s = \left. \frac{\partial z }{\partial t} \right|_s
+ = \frac{\partial z}{\partial s} \; \frac{\partial s}{\partial t}
+ = e_3 \, \frac{\partial s}{\partial t}
+\end{equation}
+
+
+% ================================================================
+% continuity equation
+% ================================================================
+\section{Continuity Equation in $s-$coordinates}
+\label{Apdx_A_continuity}
+
+Using (\ref{Apdx_A_s_chain_rule}) and the fact that the horizontal scale factors
+$e_1$ and $e_2$ do not depend on the vertical coordinate, the divergence of
+the velocity relative to the ($i$,$j$,$z$) coordinate system is transformed as follows
+in order to obtain its expression in the curvilinear $s-$coordinate system:
+
+\begin{subequations}
+\begin{align*} {\begin{array}{*{20}l}
+\nabla \cdot {\rm {\bf U}}
+&= \frac{1}{e_1 \,e_2 } \left[ \left. {\frac{\partial (e_2 \,u)}{\partial i}} \right|_z
+ +\left. {\frac{\partial(e_1 \,v)}{\partial j}} \right|_z \right]
++ \frac{\partial w}{\partial z} \\
+\\
+& = \frac{1}{e_1 \,e_2 } \left[
+ \left. \frac{\partial (e_2 \,u)}{\partial i} \right|_s
+ - \frac{e_1 }{e_3 } \sigma _1 \frac{\partial (e_2 \,u)}{\partial s}
+ + \left. \frac{\partial (e_1 \,v)}{\partial j} \right|_s
+ - \frac{e_2 }{e_3 } \sigma _2 \frac{\partial (e_1 \,v)}{\partial s} \right]
+ + \frac{\partial w}{\partial s} \; \frac{\partial s}{\partial z} \\
+\\
+& = \frac{1}{e_1 \,e_2 } \left[
+ \left. \frac{\partial (e_2 \,u)}{\partial i} \right|_s
+ + \left. \frac{\partial (e_1 \,v)}{\partial j} \right|_s \right]
+ + \frac{1}{e_3 }\left[ \frac{\partial w}{\partial s}
+ - \sigma _1 \frac{\partial u}{\partial s}
+ - \sigma _2 \frac{\partial v}{\partial s} \right] \\
+\\
+& = \frac{1}{e_1 \,e_2 \,e_3 } \left[
+ \left. \frac{\partial (e_2 \,e_3 \,u)}{\partial i} \right|_s
+ -\left. e_2 \,u \frac{\partial e_3 }{\partial i} \right|_s
+ + \left. \frac{\partial (e_1 \,e_3 \,v)}{\partial j} \right|_s
+ - \left. e_1 v \frac{\partial e_3 }{\partial j} \right|_s \right] \\
+& \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad
+ + \frac{1}{e_3 } \left[ \frac{\partial w}{\partial s}
+ - \sigma _1 \frac{\partial u}{\partial s}
+ - \sigma _2 \frac{\partial v}{\partial s} \right] \\
+%
+\intertext{Noting that $
+ \frac{1}{e_1} \left.{ \frac{\partial e_3}{\partial i}} \right|_s
+=\frac{1}{e_1} \left.{ \frac{\partial^2 z}{\partial i\,\partial s}} \right|_s
+=\frac{\partial}{\partial s} \left( {\frac{1}{e_1 } \left.{ \frac{\partial z}{\partial i} }\right|_s } \right)
+=\frac{\partial \sigma _1}{\partial s}
+$ and $
+\frac{1}{e_2 }\left. {\frac{\partial e_3 }{\partial j}} \right|_s
+=\frac{\partial \sigma _2}{\partial s}
+$, it becomes:}
+%
+\nabla \cdot {\rm {\bf U}}
+& = \frac{1}{e_1 \,e_2 \,e_3 } \left[
+ \left. \frac{\partial (e_2 \,e_3 \,u)}{\partial i} \right|_s
+ +\left. \frac{\partial (e_1 \,e_3 \,v)}{\partial j} \right|_s \right] \\
+& \qquad \qquad \qquad \qquad \quad
+ +\frac{1}{e_3 }\left[ {\frac{\partial w}{\partial s}-u\frac{\partial \sigma _1 }{\partial s}-v\frac{\partial \sigma _2 }{\partial s}-\sigma _1 \frac{\partial u}{\partial s}-\sigma _2 \frac{\partial v}{\partial s}} \right] \\
+\\
+& = \frac{1}{e_1 \,e_2 \,e_3 } \left[
+ \left. \frac{\partial (e_2 \,e_3 \,u)}{\partial i} \right|_s
+ +\left. \frac{\partial (e_1 \,e_3 \,v)}{\partial j} \right|_s \right]
+ + \frac{1}{e_3 } \; \frac{\partial}{\partial s} \left[ w - u\;\sigma _1 - v\;\sigma _2 \right]
+\end{array} }
+\end{align*}
+\end{subequations}
+
+Here, $w$ is the vertical velocity relative to the $z-$coordinate system.
+Introducing the dia-surface velocity component, $\omega $, defined as
+the volume flux across the moving $s$-surfaces per unit horizontal area:
+\begin{equation} \label{Apdx_A_w_s}
+\omega = w - w_s - \sigma _1 \,u - \sigma _2 \,v \\
+\end{equation}
+with $w_s$ given by \eqref{Apdx_A_w_in_s}, we obtain the expression for
+the divergence of the velocity in the curvilinear $s-$coordinate system:
+\begin{subequations}
+\begin{align*} {\begin{array}{*{20}l}
+\nabla \cdot {\rm {\bf U}}
+&= \frac{1}{e_1 \,e_2 \,e_3 } \left[
+ \left. \frac{\partial (e_2 \,e_3 \,u)}{\partial i} \right|_s
+ +\left. \frac{\partial (e_1 \,e_3 \,v)}{\partial j} \right|_s \right]
++ \frac{1}{e_3 } \frac{\partial \omega }{\partial s}
++ \frac{1}{e_3 } \frac{\partial w_s }{\partial s} \\
+\\
+&= \frac{1}{e_1 \,e_2 \,e_3 } \left[
+ \left. \frac{\partial (e_2 \,e_3 \,u)}{\partial i} \right|_s
+ +\left. \frac{\partial (e_1 \,e_3 \,v)}{\partial j} \right|_s \right]
++ \frac{1}{e_3 } \frac{\partial \omega }{\partial s}
++ \frac{1}{e_3 } \frac{\partial}{\partial s} \left( e_3 \; \frac{\partial s}{\partial t} \right) \\
+\\
+&= \frac{1}{e_1 \,e_2 \,e_3 } \left[
+ \left. \frac{\partial (e_2 \,e_3 \,u)}{\partial i} \right|_s
+ +\left. \frac{\partial (e_1 \,e_3 \,v)}{\partial j} \right|_s \right]
++ \frac{1}{e_3 } \frac{\partial \omega }{\partial s}
++ \frac{\partial}{\partial s} \frac{\partial s}{\partial t}
++ \frac{1}{e_3 } \frac{\partial s}{\partial t} \frac{\partial e_3}{\partial s} \\
+\\
+&= \frac{1}{e_1 \,e_2 \,e_3 } \left[
+ \left. \frac{\partial (e_2 \,e_3 \,u)}{\partial i} \right|_s
+ +\left. \frac{\partial (e_1 \,e_3 \,v)}{\partial j} \right|_s \right]
++ \frac{1}{e_3 } \frac{\partial \omega }{\partial s}
++ \frac{1}{e_3 } \frac{\partial e_3}{\partial t} \\
+\end{array} }
+\end{align*}
+\end{subequations}
+
+As a result, the continuity equation \eqref{Eq_PE_continuity} in the
+$s-$coordinates is:
+\begin{equation} \label{Apdx_A_sco_Continuity}
+\frac{1}{e_3 } \frac{\partial e_3}{\partial t}
++ \frac{1}{e_1 \,e_2 \,e_3 }\left[
+ {\left. {\frac{\partial (e_2 \,e_3 \,u)}{\partial i}} \right|_s
+ + \left. {\frac{\partial (e_1 \,e_3 \,v)}{\partial j}} \right|_s } \right]
+ +\frac{1}{e_3 }\frac{\partial \omega }{\partial s} = 0
+\end{equation}
+A additional term has appeared that take into account the contribution of the time variation
+of the vertical coordinate to the volume budget.
+
+
+% ================================================================
+% momentum equation
+% ================================================================
+\section{Momentum Equation in $s-$coordinate}
+\label{Apdx_A_momentum}
+
+Here we only consider the first component of the momentum equation,
+the generalization to the second one being straightforward.
+
+$\ $\newline % force a new ligne
+
+$\bullet$ \textbf{Total derivative in vector invariant form}
+
+Let us consider \eqref{Eq_PE_dyn_vect}, the first component of the momentum
+equation in the vector invariant form. Its total $z-$coordinate time derivative,
+$\left. \frac{D u}{D t} \right|_z$ can be transformed as follows in order to obtain
+its expression in the curvilinear $s-$coordinate system:
+
+\begin{subequations}
+\begin{align*} {\begin{array}{*{20}l}
+\left. \frac{D u}{D t} \right|_z
+&= \left. {\frac{\partial u }{\partial t}} \right|_z
+ - \left. \zeta \right|_z v
+ + \frac{1}{2e_1} \left.{ \frac{\partial (u^2+v^2)}{\partial i}} \right|_z
+ + w \;\frac{\partial u}{\partial z} \\
+\\
+&= \left. {\frac{\partial u }{\partial t}} \right|_z
+ - \left. \zeta \right|_z v
+ + \frac{1}{e_1 \,e_2 }\left[ { \left.{ \frac{\partial (e_2 \,v)}{\partial i} }\right|_z
+ -\left.{ \frac{\partial (e_1 \,u)}{\partial j} }\right|_z } \right] \; v
+ + \frac{1}{2e_1} \left.{ \frac{\partial (u^2+v^2)}{\partial i} } \right|_z
+ + w \;\frac{\partial u}{\partial z} \\
+%
+\intertext{introducing the chain rule (\ref{Apdx_A_s_chain_rule}) }
+%
+&= \left. {\frac{\partial u }{\partial t}} \right|_z
+ - \frac{1}{e_1\,e_2}\left[ { \left.{ \frac{\partial (e_2 \,v)}{\partial i} } \right|_s
+ -\left.{ \frac{\partial (e_1 \,u)}{\partial j} } \right|_s } \right.
+ \left. {-\frac{e_1}{e_3}\sigma _1 \frac{\partial (e_2 \,v)}{\partial s}
+ +\frac{e_2}{e_3}\sigma _2 \frac{\partial (e_1 \,u)}{\partial s}} \right] \; v \\
+& \qquad \qquad \qquad \qquad
+ { + \frac{1}{2e_1} \left( \left. \frac{\partial (u^2+v^2)}{\partial i} \right|_s
+ - \frac{e_1}{e_3}\sigma _1 \frac{\partial (u^2+v^2)}{\partial s} \right)
+ + \frac{w}{e_3 } \;\frac{\partial u}{\partial s} } \\
+\\
+&= \left. {\frac{\partial u }{\partial t}} \right|_z
+ + \left. \zeta \right|_s \;v
+ + \frac{1}{2\,e_1}\left. {\frac{\partial (u^2+v^2)}{\partial i}} \right|_s \\
+&\qquad \qquad \qquad \quad
+ + \frac{w}{e_3 } \;\frac{\partial u}{\partial s}
+ - \left[ {\frac{\sigma _1 }{e_3 }\frac{\partial v}{\partial s}
+ - \frac{\sigma_2 }{e_3 }\frac{\partial u}{\partial s}} \right]\;v
+ - \frac{\sigma _1 }{2e_3 }\frac{\partial (u^2+v^2)}{\partial s} \\
+\\
+&= \left. {\frac{\partial u }{\partial t}} \right|_z
+ + \left. \zeta \right|_s \;v
+ + \frac{1}{2\,e_1}\left. {\frac{\partial (u^2+v^2)}{\partial i}} \right|_s \\
+&\qquad \qquad \qquad \quad
+ + \frac{1}{e_3} \left[ {w\frac{\partial u}{\partial s}
+ +\sigma _1 v\frac{\partial v}{\partial s} - \sigma _2 v\frac{\partial u}{\partial s}
+ - \sigma _1 u\frac{\partial u}{\partial s} - \sigma _1 v\frac{\partial v}{\partial s}} \right] \\
+\\
+&= \left. {\frac{\partial u }{\partial t}} \right|_z
+ + \left. \zeta \right|_s \;v
+ + \frac{1}{2\,e_1}\left. {\frac{\partial (u^2+v^2)}{\partial i}} \right|_s
+ + \frac{1}{e_3} \left[ w - \sigma _2 v - \sigma _1 u \right]
+ \; \frac{\partial u}{\partial s} \\
+%
+\intertext{Introducing $\omega$, the dia-a-surface velocity given by (\ref{Apdx_A_w_s}) }
+%
+&= \left. {\frac{\partial u }{\partial t}} \right|_z
+ + \left. \zeta \right|_s \;v
+ + \frac{1}{2\,e_1}\left. {\frac{\partial (u^2+v^2)}{\partial i}} \right|_s
+ + \frac{1}{e_3 } \left( \omega - w_s \right) \frac{\partial u}{\partial s} \\
+\end{array} }
+\end{align*}
+\end{subequations}
+%
+Applying the time derivative chain rule (first equation of (\ref{Apdx_A_s_chain_rule}))
+to $u$ and using (\ref{Apdx_A_w_in_s}) provides the expression of the last term
+of the right hand side,
+\begin{equation*} {\begin{array}{*{20}l}
+w_s \;\frac{\partial u}{\partial s}
+ = \frac{\partial s}{\partial t} \; \frac{\partial u }{\partial s}
+ = \left. {\frac{\partial u }{\partial t}} \right|_s - \left. {\frac{\partial u }{\partial t}} \right|_z \quad ,
+\end{array} }
+\end{equation*}
+leads to the $s-$coordinate formulation of the total $z-$coordinate time derivative,
+$i.e.$ the total $s-$coordinate time derivative :
+\begin{align} \label{Apdx_A_sco_Dt_vect}
+\left. \frac{D u}{D t} \right|_s
+ = \left. {\frac{\partial u }{\partial t}} \right|_s
+ + \left. \zeta \right|_s \;v
+ + \frac{1}{2\,e_1}\left. {\frac{\partial (u^2+v^2)}{\partial i}} \right|_s
+ + \frac{1}{e_3 } \omega \;\frac{\partial u}{\partial s}
+\end{align}
+Therefore, the vector invariant form of the total time derivative has exactly the same
+mathematical form in $z-$ and $s-$coordinates. This is not the case for the flux form
+as shown in next paragraph.
+
+$\ $\newline % force a new ligne
+
+$\bullet$ \textbf{Total derivative in flux form}
+
+Let us start from the total time derivative in the curvilinear $s-$coordinate system
+we have just establish. Following the procedure used to establish (\ref{Eq_PE_flux_form}),
+it can be transformed into :
+%\begin{subequations}
+\begin{align*} {\begin{array}{*{20}l}
+\left. \frac{D u}{D t} \right|_s &= \left. {\frac{\partial u }{\partial t}} \right|_s
+ & - \zeta \;v
+ + \frac{1}{2\;e_1 } \frac{\partial \left( {u^2+v^2} \right)}{\partial i}
+ + \frac{1}{e_3} \omega \;\frac{\partial u}{\partial s} \\
+\\
+ &= \left. {\frac{\partial u }{\partial t}} \right|_s
+ &+\frac{1}{e_1\;e_2} \left( \frac{\partial \left( {e_2 \,u\,u } \right)}{\partial i}
+ + \frac{\partial \left( {e_1 \,u\,v } \right)}{\partial j} \right)
+ + \frac{1}{e_3 } \frac{\partial \left( {\omega\,u} \right)}{\partial s} \\
+\\
+ &&- \,u \left[ \frac{1}{e_1 e_2 } \left( \frac{\partial(e_2 u)}{\partial i}
+ + \frac{\partial(e_1 v)}{\partial j} \right)
+ + \frac{1}{e_3} \frac{\partial \omega}{\partial s} \right] \\
+\\
+ &&- \frac{v}{e_1 e_2 }\left( v \;\frac{\partial e_2 }{\partial i}
+ -u \;\frac{\partial e_1 }{\partial j} \right) \\
+\end{array} }
+\end{align*}
+%
+Introducing the vertical scale factor inside the horizontal derivative of the first two terms
+($i.e.$ the horizontal divergence), it becomes :
+\begin{subequations}
+\begin{align*} {\begin{array}{*{20}l}
+%\begin{align*} {\begin{array}{*{20}l}
+%{\begin{array}{*{20}l}
+\left. \frac{D u}{D t} \right|_s
+ &= \left. {\frac{\partial u }{\partial t}} \right|_s
+ &+ \frac{1}{e_1\,e_2\,e_3} \left( \frac{\partial( e_2 e_3 \,u^2 )}{\partial i}
+ + \frac{\partial( e_1 e_3 \,u v )}{\partial j}
+ - e_2 u u \frac{\partial e_3}{\partial i}
+ - e_1 u v \frac{\partial e_3 }{\partial j} \right)
+ + \frac{1}{e_3} \frac{\partial \left( {\omega\,u} \right)}{\partial s} \\
+\\
+ && - \,u \left[ \frac{1}{e_1 e_2 e_3} \left( \frac{\partial(e_2 e_3 \, u)}{\partial i}
+ + \frac{\partial(e_1 e_3 \, v)}{\partial j}
+ - e_2 u \;\frac{\partial e_3 }{\partial i}
+ - e_1 v \;\frac{\partial e_3 }{\partial j} \right)
+ -\frac{1}{e_3} \frac{\partial \omega}{\partial s} \right] \\
+\\
+ && - \frac{v}{e_1 e_2 }\left( v \;\frac{\partial e_2 }{\partial i}
+ -u \;\frac{\partial e_1 }{\partial j} \right) \\
+\\
+ &= \left. {\frac{\partial u }{\partial t}} \right|_s
+ &+ \frac{1}{e_1\,e_2\,e_3} \left( \frac{\partial( e_2 e_3 \,u\,u )}{\partial i}
+ + \frac{\partial( e_1 e_3 \,u\,v )}{\partial j} \right)
+ + \frac{1}{e_3 } \frac{\partial \left( {\omega\,u} \right)}{\partial s} \\
+\\
+&& - \,u \left[ \frac{1}{e_1 e_2 e_3} \left( \frac{\partial(e_2 e_3 \, u)}{\partial i}
+ + \frac{\partial(e_1 e_3 \, v)}{\partial j} \right)
+ -\frac{1}{e_3} \frac{\partial \omega}{\partial s} \right]
+ - \frac{v}{e_1 e_2 }\left( v \;\frac{\partial e_2 }{\partial i}
+ -u \;\frac{\partial e_1 }{\partial j} \right) \\
+%
+\intertext {Introducing a more compact form for the divergence of the momentum fluxes,
+and using (\ref{Apdx_A_sco_Continuity}), the $s-$coordinate continuity equation,
+it becomes : }
+%
+ &= \left. {\frac{\partial u }{\partial t}} \right|_s
+ &+ \left. \nabla \cdot \left( {{\rm {\bf U}}\,u} \right) \right|_s
+ + \,u \frac{1}{e_3 } \frac{\partial e_3}{\partial t}
+ - \frac{v}{e_1 e_2 }\left( v \;\frac{\partial e_2 }{\partial i}
+ -u \;\frac{\partial e_1 }{\partial j} \right) \\
+\end{array} }
+\end{align*}
+\end{subequations}
+which leads to the $s-$coordinate flux formulation of the total $s-$coordinate time derivative,
+$i.e.$ the total $s-$coordinate time derivative in flux form :
+\begin{flalign}\label{Apdx_A_sco_Dt_flux}
+\left. \frac{D u}{D t} \right|_s = \frac{1}{e_3} \left. \frac{\partial ( e_3\,u)}{\partial t} \right|_s
+ + \left. \nabla \cdot \left( {{\rm {\bf U}}\,u} \right) \right|_s
+ - \frac{v}{e_1 e_2 }\left( v \;\frac{\partial e_2 }{\partial i}
+ -u \;\frac{\partial e_1 }{\partial j} \right)
+\end{flalign}
+which is the total time derivative expressed in the curvilinear $s-$coordinate system.
+It has the same form as in the $z-$coordinate but for the vertical scale factor
+that has appeared inside the time derivative which comes from the modification
+of (\ref{Apdx_A_sco_Continuity}), the continuity equation.
+
+$\ $\newline % force a new ligne
+
+$\bullet$ \textbf{horizontal pressure gradient}
+
+The horizontal pressure gradient term can be transformed as follows:
+\begin{equation*}
+\begin{split}
+ -\frac{1}{\rho _o \, e_1 }\left. {\frac{\partial p}{\partial i}} \right|_z
+ & =-\frac{1}{\rho _o e_1 }\left[ {\left. {\frac{\partial p}{\partial i}} \right|_s -\frac{e_1 }{e_3 }\sigma _1 \frac{\partial p}{\partial s}} \right] \\
+& =-\frac{1}{\rho _o \,e_1 }\left. {\frac{\partial p}{\partial i}} \right|_s +\frac{\sigma _1 }{\rho _o \,e_3 }\left( {-g\;\rho \;e_3 } \right) \\
+&=-\frac{1}{\rho _o \,e_1 }\left. {\frac{\partial p}{\partial i}} \right|_s -\frac{g\;\rho }{\rho _o }\sigma _1
+\end{split}
+\end{equation*}
+Applying similar manipulation to the second component and replacing
+$\sigma _1$ and $\sigma _2$ by their expression \eqref{Apdx_A_s_slope}, it comes:
+\begin{equation} \label{Apdx_A_grad_p}
+\begin{split}
+ -\frac{1}{\rho _o \, e_1 } \left. {\frac{\partial p}{\partial i}} \right|_z
+&=-\frac{1}{\rho _o \,e_1 } \left( \left. {\frac{\partial p}{\partial i}} \right|_s
+ + g\;\rho \;\left. {\frac{\partial z}{\partial i}} \right|_s \right) \\
+%
+ -\frac{1}{\rho _o \, e_2 }\left. {\frac{\partial p}{\partial j}} \right|_z
+&=-\frac{1}{\rho _o \,e_2 } \left( \left. {\frac{\partial p}{\partial j}} \right|_s
+ + g\;\rho \;\left. {\frac{\partial z}{\partial j}} \right|_s \right) \\
+\end{split}
+\end{equation}
+
+An additional term appears in (\ref{Apdx_A_grad_p}) which accounts for the
+tilt of $s-$surfaces with respect to geopotential $z-$surfaces.
+
+As in $z$-coordinate, the horizontal pressure gradient can be split in two parts
+following \citet{Marsaleix_al_OM08}. Let defined a density anomaly, $d$, by $d=(\rho - \rho_o)/ \rho_o$,
+and a hydrostatic pressure anomaly, $p_h'$, by $p_h'= g \; \int_z^\eta d \; e_3 \; dk$.
+The pressure is then given by:
+\begin{equation*}
+\begin{split}
+p &= g\; \int_z^\eta \rho \; e_3 \; dk = g\; \int_z^\eta \left( \rho_o \, d + 1 \right) \; e_3 \; dk \\
+ &= g \, \rho_o \; \int_z^\eta d \; e_3 \; dk + g \, \int_z^\eta e_3 \; dk
+\end{split}
+\end{equation*}
+Therefore, $p$ and $p_h'$ are linked through:
+\begin{equation} \label{Apdx_A_pressure}
+ p = \rho_o \; p_h' + g \, ( z + \eta )
+\end{equation}
+and the hydrostatic pressure balance expressed in terms of $p_h'$ and $d$ is:
+\begin{equation*}
+\frac{\partial p_h'}{\partial k} = - d \, g \, e_3
+\end{equation*}
+
+Substituing \eqref{Apdx_A_pressure} in \eqref{Apdx_A_grad_p} and using the definition of
+the density anomaly it comes the expression in two parts:
+\begin{equation} \label{Apdx_A_grad_p}
+\begin{split}
+ -\frac{1}{\rho _o \, e_1 } \left. {\frac{\partial p}{\partial i}} \right|_z
+&=-\frac{1}{e_1 } \left( \left. {\frac{\partial p_h'}{\partial i}} \right|_s
+ + g\; d \;\left. {\frac{\partial z}{\partial i}} \right|_s \right) - \frac{g}{e_1 } \frac{\partial \eta}{\partial i} \\
+%
+ -\frac{1}{\rho _o \, e_2 }\left. {\frac{\partial p}{\partial j}} \right|_z
+&=-\frac{1}{e_2 } \left( \left. {\frac{\partial p_h'}{\partial j}} \right|_s
+ + g\; d \;\left. {\frac{\partial z}{\partial j}} \right|_s \right) - \frac{g}{e_2 } \frac{\partial \eta}{\partial j}\\
+\end{split}
+\end{equation}
+This formulation of the pressure gradient is characterised by the appearance of a term depending on the
+the sea surface height only (last term on the right hand side of expression \eqref{Apdx_A_grad_p}).
+This term will be loosely termed \textit{surface pressure gradient}
+whereas the first term will be termed the
+\textit{hydrostatic pressure gradient} by analogy to the $z$-coordinate formulation.
+In fact, the the true surface pressure gradient is $1/\rho_o \nabla (\rho \eta)$, and
+$\eta$ is implicitly included in the computation of $p_h'$ through the upper bound of
+the vertical integration.
+
+
+$\ $\newline % force a new ligne
+
+$\bullet$ \textbf{The other terms of the momentum equation}
+
+The coriolis and forcing terms as well as the the vertical physics remain unchanged
+as they involve neither time nor space derivatives. The form of the lateral physics is
+discussed in appendix~\ref{Apdx_B}.
+
+
+$\ $\newline % force a new ligne
+
+$\bullet$ \textbf{Full momentum equation}
+
+To sum up, in a curvilinear $s$-coordinate system, the vector invariant momentum equation
+solved by the model has the same mathematical expression as the one in a curvilinear
+$z-$coordinate, except for the pressure gradient term :
+\begin{subequations} \label{Apdx_A_dyn_vect}
+\begin{multline} \label{Apdx_A_PE_dyn_vect_u}
+ \frac{\partial u}{\partial t}=
+ + \left( {\zeta +f} \right)\,v
+ - \frac{1}{2\,e_1} \frac{\partial}{\partial i} \left( u^2+v^2 \right)
+ - \frac{1}{e_3} \omega \frac{\partial u}{\partial k} \\
+ - \frac{1}{e_1 } \left( \frac{\partial p_h'}{\partial i} + g\; d \; \frac{\partial z}{\partial i} \right)
+ - \frac{g}{e_1 } \frac{\partial \eta}{\partial i}
+ + D_u^{\vect{U}} + F_u^{\vect{U}}
+\end{multline}
+\begin{multline} \label{Apdx_A_dyn_vect_v}
+\frac{\partial v}{\partial t}=
+ - \left( {\zeta +f} \right)\,u
+ - \frac{1}{2\,e_2 }\frac{\partial }{\partial j}\left( u^2+v^2 \right)
+ - \frac{1}{e_3 } \omega \frac{\partial v}{\partial k} \\
+ - \frac{1}{e_2 } \left( \frac{\partial p_h'}{\partial j} + g\; d \; \frac{\partial z}{\partial j} \right)
+ - \frac{g}{e_2 } \frac{\partial \eta}{\partial j}
+ + D_v^{\vect{U}} + F_v^{\vect{U}}
+\end{multline}
+\end{subequations}
+whereas the flux form momentum equation differ from it by the formulation of both
+the time derivative and the pressure gradient term :
+\begin{subequations} \label{Apdx_A_dyn_flux}
+\begin{multline} \label{Apdx_A_PE_dyn_flux_u}
+ \frac{1}{e_3} \frac{\partial \left( e_3\,u \right) }{\partial t} =
+ \nabla \cdot \left( {{\rm {\bf U}}\,u} \right)
+ + \left\{ {f + \frac{1}{e_1 e_2 }\left( v \;\frac{\partial e_2 }{\partial i}
+ -u \;\frac{\partial e_1 }{\partial j} \right)} \right\} \,v \\
+ - \frac{1}{e_1 } \left( \frac{\partial p_h'}{\partial i} + g\; d \; \frac{\partial z}{\partial i} \right)
+ - \frac{g}{e_1 } \frac{\partial \eta}{\partial i}
+ + D_u^{\vect{U}} + F_u^{\vect{U}}
+\end{multline}
+\begin{multline} \label{Apdx_A_dyn_flux_v}
+ \frac{1}{e_3}\frac{\partial \left( e_3\,v \right) }{\partial t}=
+ - \nabla \cdot \left( {{\rm {\bf U}}\,v} \right)
+ + \left\{ {f + \frac{1}{e_1 e_2 }\left( v \;\frac{\partial e_2 }{\partial i}
+ -u \;\frac{\partial e_1 }{\partial j} \right)} \right\} \,u \\
+ - \frac{1}{e_2 } \left( \frac{\partial p_h'}{\partial j} + g\; d \; \frac{\partial z}{\partial j} \right)
+ - \frac{g}{e_2 } \frac{\partial \eta}{\partial j}
+ + D_v^{\vect{U}} + F_v^{\vect{U}}
+\end{multline}
+\end{subequations}
+Both formulation share the same hydrostatic pressure balance expressed in terms of
+hydrostatic pressure and density anomalies, $p_h'$ and $d=( \frac{\rho}{\rho_o}-1 )$:
+\begin{equation} \label{Apdx_A_dyn_zph}
+\frac{\partial p_h'}{\partial k} = - d \, g \, e_3
+\end{equation}
+
+It is important to realize that the change in coordinate system has only concerned
+the position on the vertical. It has not affected (\textbf{i},\textbf{j},\textbf{k}), the
+orthogonal curvilinear set of unit vectors. ($u$,$v$) are always horizontal velocities
+so that their evolution is driven by \emph{horizontal} forces, in particular
+the pressure gradient. By contrast, $\omega$ is not $w$, the third component of the velocity,
+but the dia-surface velocity component, $i.e.$ the volume flux across the moving
+$s$-surfaces per unit horizontal area.
+
+
+% ================================================================
+% Tracer equation
+% ================================================================
+\section{Tracer Equation}
+\label{Apdx_A_tracer}
+
+The tracer equation is obtained using the same calculation as for the continuity
+equation and then regrouping the time derivative terms in the left hand side :
+
+\begin{multline} \label{Apdx_A_tracer}
+ \frac{1}{e_3} \frac{\partial \left( e_3 T \right)}{\partial t}
+ = -\frac{1}{e_1 \,e_2 \,e_3}
+ \left[ \frac{\partial }{\partial i} \left( {e_2 \,e_3 \;Tu} \right)
+ + \frac{\partial }{\partial j} \left( {e_1 \,e_3 \;Tv} \right) \right] \\
+ + \frac{1}{e_3} \frac{\partial }{\partial k} \left( Tw \right)
+ + D^{T} +F^{T}
+\end{multline}
+
+
+The expression for the advection term is a straight consequence of (A.4), the
+expression of the 3D divergence in the $s-$coordinates established above.
+
Index: /branches/UKMO/dev_r5518_GO6_under_ice_relax_dr_hook/DOC/TexFiles/Chapters/Annex_B.tex
===================================================================
--- /branches/UKMO/dev_r5518_GO6_under_ice_relax_dr_hook/DOC/TexFiles/Chapters/Annex_B.tex (revision 11643)
+++ /branches/UKMO/dev_r5518_GO6_under_ice_relax_dr_hook/DOC/TexFiles/Chapters/Annex_B.tex (revision 11643)
@@ -0,0 +1,365 @@
+% ================================================================
+% Chapter Ñ Appendix B : Diffusive Operators
+% ================================================================
+\chapter{Appendix B : Diffusive Operators}
+\label{Apdx_B}
+\minitoc
+
+
+\newpage
+$\ $\newline % force a new ligne
+
+% ================================================================
+% Horizontal/Vertical 2nd Order Tracer Diffusive Operators
+% ================================================================
+\section{Horizontal/Vertical 2nd Order Tracer Diffusive Operators}
+\label{Apdx_B_1}
+
+\subsubsection*{In z-coordinates}
+In $z$-coordinates, the horizontal/vertical second order tracer diffusion operator
+is given by:
+\begin{eqnarray} \label{Apdx_B1}
+ &D^T = \frac{1}{e_1 \, e_2} \left[
+ \left. \frac{\partial}{\partial i} \left( \frac{e_2}{e_1}A^{lT} \;\left. \frac{\partial T}{\partial i} \right|_z \right) \right|_z \right.
+ \left.
++ \left. \frac{\partial}{\partial j} \left( \frac{e_1}{e_2}A^{lT} \;\left. \frac{\partial T}{\partial j} \right|_z \right) \right|_z \right]
++ \frac{\partial }{\partial z}\left( {A^{vT} \;\frac{\partial T}{\partial z}} \right)
+\end{eqnarray}
+
+\subsubsection*{In generalized vertical coordinates}
+In $s$-coordinates, we defined the slopes of $s$-surfaces, $\sigma_1$ and
+$\sigma_2$ by \eqref{Apdx_A_s_slope} and the vertical/horizontal ratio of diffusion
+coefficient by $\epsilon = A^{vT} / A^{lT}$. The diffusion operator is given by:
+
+\begin{equation} \label{Apdx_B2}
+D^T = \left. \nabla \right|_s \cdot
+ \left[ A^{lT} \;\Re \cdot \left. \nabla \right|_s T \right] \\
+\;\;\text{where} \;\Re =\left( {{\begin{array}{*{20}c}
+ 1 \hfill & 0 \hfill & {-\sigma _1 } \hfill \\
+ 0 \hfill & 1 \hfill & {-\sigma _2 } \hfill \\
+ {-\sigma _1 } \hfill & {-\sigma _2 } \hfill & {\varepsilon +\sigma _1
+^2+\sigma _2 ^2} \hfill \\
+\end{array} }} \right)
+\end{equation}
+or in expanded form:
+\begin{subequations}
+\begin{align*} {\begin{array}{*{20}l}
+D^T=& \frac{1}{e_1\,e_2\,e_3 }\;\left[ {\ \ \ \ e_2\,e_3\,A^{lT} \;\left.
+{\frac{\partial }{\partial i}\left( {\frac{1}{e_1}\;\left. {\frac{\partial T}{\partial i}} \right|_s -\frac{\sigma _1 }{e_3 }\;\frac{\partial T}{\partial s}} \right)} \right|_s } \right. \\
+&\qquad \quad \ \ \ +e_1\,e_3\,A^{lT} \;\left. {\frac{\partial }{\partial j}\left( {\frac{1}{e_2 }\;\left. {\frac{\partial T}{\partial j}} \right|_s -\frac{\sigma _2 }{e_3 }\;\frac{\partial T}{\partial s}} \right)} \right|_s \\
+&\qquad \quad \ \ \ + e_1\,e_2\,A^{lT} \;\frac{\partial }{\partial s}\left( {-\frac{\sigma _1 }{e_1 }\;\left. {\frac{\partial T}{\partial i}} \right|_s -\frac{\sigma _2 }{e_2 }\;\left. {\frac{\partial T}{\partial j}} \right|_s } \right.
+ \left. {\left. {+\left( {\varepsilon +\sigma _1^2+\sigma _2 ^2} \right)\;\frac{1}{e_3 }\;\frac{\partial T}{\partial s}} \right)\;\;} \right]
+\end{array} }
+\end{align*}
+\end{subequations}
+
+Equation \eqref{Apdx_B2} is obtained from \eqref{Apdx_B1} without any
+additional assumption. Indeed, for the special case $k=z$ and thus $e_3 =1$,
+we introduce an arbitrary vertical coordinate $s = s (i,j,z)$ as in Appendix~\ref{Apdx_A}
+and use \eqref{Apdx_A_s_slope} and \eqref{Apdx_A_s_chain_rule}.
+Since no cross horizontal derivative $\partial _i \partial _j $ appears in
+\eqref{Apdx_B1}, the ($i$,$z$) and ($j$,$z$) planes are independent.
+The derivation can then be demonstrated for the ($i$,$z$)~$\to$~($j$,$s$)
+transformation without any loss of generality:
+
+\begin{subequations}
+\begin{align*} {\begin{array}{*{20}l}
+D^T&=\frac{1}{e_1\,e_2} \left. {\frac{\partial }{\partial i}\left( {\frac{e_2}{e_1}A^{lT}\;\left. {\frac{\partial T}{\partial i}} \right|_z } \right)} \right|_z
+ +\frac{\partial }{\partial z}\left( {A^{vT}\;\frac{\partial T}{\partial z}} \right) \\
+ \\
+%
+&=\frac{1}{e_1\,e_2 }\left[ {\left. {\;\frac{\partial }{\partial i}\left( {\frac{e_2}{e_1}A^{lT}\;\left( {\left. {\frac{\partial T}{\partial i}} \right|_s
+ -\frac{e_1\,\sigma _1 }{e_3 }\frac{\partial T}{\partial s}} \right)} \right)} \right|_s } \right. \\
+& \qquad \qquad \left. { -\frac{e_1\,\sigma _1 }{e_3 }\frac{\partial }{\partial s}\left( {\frac{e_2 }{e_1 }A^{lT}\;\left. {\left( {\left. {\frac{\partial T}{\partial i}} \right|_s -\frac{e_1 \,\sigma _1 }{e_3 }\frac{\partial T}{\partial s}} \right)} \right|_s } \right)\;} \right]
+\shoveright{ +\frac{1}{e_3 }\frac{\partial }{\partial s}\left[ {\frac{A^{vT}}{e_3 }\;\frac{\partial T}{\partial s}} \right]} \qquad \qquad \qquad \\
+ \\
+%
+&=\frac{1}{e_1 \,e_2 \,e_3 }\left[ {\left. {\;\;\frac{\partial }{\partial i}\left( {\frac{e_2 \,e_3 }{e_1 }A^{lT}\;\left. {\frac{\partial T}{\partial i}} \right|_s } \right)} \right|_s -\left. {\frac{e_2 }{e_1}A^{lT}\;\frac{\partial e_3 }{\partial i}} \right|_s \left. {\frac{\partial T}{\partial i}} \right|_s } \right. \\
+& \qquad \qquad \quad \left. {-e_3 \frac{\partial }{\partial i}\left( {\frac{e_2 \,\sigma _1 }{e_3 }A^{lT}\;\frac{\partial T}{\partial s}} \right)} \right|_s -e_1 \,\sigma _1 \frac{\partial }{\partial s}\left( {\frac{e_2 }{e_1 }A^{lT}\;\left. {\frac{\partial T}{\partial i}} \right|_s } \right) \\
+& \qquad \qquad \quad \shoveright{ -e_1 \,\sigma _1 \frac{\partial }{\partial s}\left( {-\frac{e_2 \,\sigma _1 }{e_3 }A^{lT}\;\frac{\partial T}{\partial s}} \right)\;\,\left. {+\frac{\partial }{\partial s}\left( {\frac{e_1 \,e_2 }{e_3 }A^{vT}\;\frac{\partial T}{\partial s}} \right)\quad} \right] }\\
+\end{array} } \\
+%
+ {\begin{array}{*{20}l}
+\intertext{Noting that $\frac{1}{e_1} \left. \frac{\partial e_3 }{\partial i} \right|_s = \frac{\partial \sigma _1 }{\partial s}$, it becomes:}
+%
+& =\frac{1}{e_1\,e_2\,e_3 }\left[ {\left. {\;\;\;\frac{\partial }{\partial i}\left( {\frac{e_2\,e_3 }{e_1}\,A^{lT}\;\left. {\frac{\partial T}{\partial i}} \right|_s } \right)} \right|_s \left. -\, {e_3 \frac{\partial }{\partial i}\left( {\frac{e_2 \,\sigma _1 }{e_3 }A^{lT}\;\frac{\partial T}{\partial s}} \right)} \right|_s } \right. \\
+& \qquad \qquad \quad -e_2 A^{lT}\;\frac{\partial \sigma _1 }{\partial s}\left. {\frac{\partial T}{\partial i}} \right|_s -e_1 \,\sigma_1 \frac{\partial }{\partial s}\left( {\frac{e_2 }{e_1 }A^{lT}\;\left. {\frac{\partial T}{\partial i}} \right|_s } \right) \\
+& \qquad \qquad \quad\shoveright{ \left. { +e_1 \,\sigma _1 \frac{\partial }{\partial s}\left( {\frac{e_2 \,\sigma _1 }{e_3 }A^{lT}\;\frac{\partial T}{\partial s}} \right)+\frac{\partial }{\partial s}\left( {\frac{e_1 \,e_2 }{e_3 }A^{vT}\;\frac{\partial T}{\partial z}} \right)\;\;\;} \right] }\\
+\\
+&=\frac{1}{e_1 \,e_2 \,e_3 } \left[ {\left. {\;\;\;\frac{\partial }{\partial i} \left( {\frac{e_2 \,e_3 }{e_1 }A^{lT}\;\left. {\frac{\partial T}{\partial i}} \right|_s } \right)} \right|_s \left. {-\frac{\partial }{\partial i}\left( {e_2 \,\sigma _1 A^{lT}\;\frac{\partial T}{\partial s}} \right)} \right|_s } \right. \\
+& \qquad \qquad \quad \left. {+\frac{e_2 \,\sigma _1 }{e_3}A^{lT}\;\frac{\partial T}{\partial s} \;\frac{\partial e_3 }{\partial i}} \right|_s -e_2 A^{lT}\;\frac{\partial \sigma _1 }{\partial s}\left. {\frac{\partial T}{\partial i}} \right|_s \\
+& \qquad \qquad \quad-e_2 \,\sigma _1 \frac{\partial}{\partial s}\left( {A^{lT}\;\left. {\frac{\partial T}{\partial i}} \right|_s } \right)+\frac{\partial }{\partial s}\left( {\frac{e_1 \,e_2 \,\sigma _1 ^2}{e_3 }A^{lT}\;\frac{\partial T}{\partial s}} \right) \\
+& \qquad \qquad \quad\shoveright{ \left. {-\frac{\partial \left( {e_1 \,e_2 \,\sigma _1 } \right)}{\partial s} \left( {\frac{\sigma _1 }{e_3}A^{lT}\;\frac{\partial T}{\partial s}} \right) + \frac{\partial }{\partial s}\left( {\frac{e_1 \,e_2 }{e_3 }A^{vT}\;\frac{\partial T}{\partial s}} \right)\;\;\;} \right]}
+\end{array} } \\
+{\begin{array}{*{20}l}
+%
+\intertext{using the same remark as just above, it becomes:}
+%
+&= \frac{1}{e_1 \,e_2 \,e_3 } \left[ {\left. {\;\;\;\frac{\partial }{\partial i} \left( {\frac{e_2 \,e_3 }{e_1 }A^{lT}\;\left. {\frac{\partial T}{\partial i}} \right|_s -e_2 \,\sigma _1 A^{lT}\;\frac{\partial T}{\partial s}} \right)} \right|_s } \right.\;\;\; \\
+& \qquad \qquad \quad+\frac{e_1 \,e_2 \,\sigma _1 }{e_3 }A^{lT}\;\frac{\partial T}{\partial s}\;\frac{\partial \sigma _1 }{\partial s} - \frac {\sigma _1 }{e_3} A^{lT} \;\frac{\partial \left( {e_1 \,e_2 \,\sigma _1 } \right)}{\partial s}\;\frac{\partial T}{\partial s} \\
+& \qquad \qquad \quad-e_2 \left( {A^{lT}\;\frac{\partial \sigma _1 }{\partial s}\left. {\frac{\partial T}{\partial i}} \right|_s +\frac{\partial }{\partial s}\left( {\sigma _1 A^{lT}\;\left. {\frac{\partial T}{\partial i}} \right|_s } \right)-\frac{\partial \sigma _1 }{\partial s}\;A^{lT}\;\left. {\frac{\partial T}{\partial i}} \right|_s } \right) \\
+& \qquad \qquad \quad\shoveright{\left. {+\frac{\partial }{\partial s}\left( {\frac{e_1 \,e_2 \,\sigma _1 ^2}{e_3 }A^{lT}\;\frac{\partial T}{\partial s}+\frac{e_1 \,e_2}{e_3 }A^{vT}\;\frac{\partial T}{\partial s}} \right)\;\;\;} \right] }
+ \end{array} } \\
+{\begin{array}{*{20}l}
+%
+\intertext{Since the horizontal scale factors do not depend on the vertical coordinate,
+the last term of the first line and the first term of the last line cancel, while
+the second line reduces to a single vertical derivative, so it becomes:}
+%
+& =\frac{1}{e_1 \,e_2 \,e_3 }\left[ {\left. {\;\;\;\frac{\partial }{\partial i}\left( {\frac{e_2 \,e_3 }{e_1 }A^{lT}\;\left. {\frac{\partial T}{\partial i}} \right|_s -e_2 \,\sigma _1 \,A^{lT}\;\frac{\partial T}{\partial s}} \right)} \right|_s } \right. \\
+& \qquad \qquad \quad \shoveright{ \left. {+\frac{\partial }{\partial s}\left( {-e_2 \,\sigma _1 \,A^{lT}\;\left. {\frac{\partial T}{\partial i}} \right|_s +A^{lT}\frac{e_1 \,e_2 }{e_3 }\;\left( {\varepsilon +\sigma _1 ^2} \right)\frac{\partial T}{\partial s}} \right)\;\;\;} \right]}
+ \\
+%
+\intertext{in other words, the horizontal/vertical Laplacian operator in the ($i$,$s$) plane takes the following form:}
+\end{array} } \\
+%
+{\frac{1}{e_1\,e_2\,e_3}}
+\left( {{\begin{array}{*{30}c}
+{\left. {\frac{\partial \left( {e_2 e_3 \bullet } \right)}{\partial i}} \right|_s } \hfill \\
+{\frac{\partial \left( {e_1 e_2 \bullet } \right)}{\partial s}} \hfill \\
+\end{array}}}\right)
+\cdot \left[ {A^{lT}
+\left( {{\begin{array}{*{30}c}
+ {1} \hfill & {-\sigma_1 } \hfill \\
+ {-\sigma_1} \hfill & {\varepsilon + \sigma_1^2} \hfill \\
+\end{array} }} \right)
+\cdot
+\left( {{\begin{array}{*{30}c}
+{\frac{1}{e_1 }\;\left. {\frac{\partial \bullet }{\partial i}} \right|_s } \hfill \\
+{\frac{1}{e_3 }\;\frac{\partial \bullet }{\partial s}} \hfill \\
+\end{array}}} \right) \left( T \right)} \right]
+\end{align*}
+\end{subequations}
+\addtocounter{equation}{-2}
+
+% ================================================================
+% Isopycnal/Vertical 2nd Order Tracer Diffusive Operators
+% ================================================================
+\section{Iso/diapycnal 2nd Order Tracer Diffusive Operators}
+\label{Apdx_B_2}
+
+\subsubsection*{In z-coordinates}
+
+The iso/diapycnal diffusive tensor $\textbf {A}_{\textbf I}$ expressed in the ($i$,$j$,$k$)
+curvilinear coordinate system in which the equations of the ocean circulation model are
+formulated, takes the following form \citep{Redi_JPO82}:
+
+\begin{equation} \label{Apdx_B3}
+\textbf {A}_{\textbf I} = \frac{A^{lT}}{\left( {1+a_1 ^2+a_2 ^2} \right)}
+\left[ {{\begin{array}{*{20}c}
+ {1+a_1 ^2} \hfill & {-a_1 a_2 } \hfill & {-a_1 } \hfill \\
+ {-a_1 a_2 } \hfill & {1+a_2 ^2} \hfill & {-a_2 } \hfill \\
+ {-a_1 } \hfill & {-a_2 } \hfill & {\varepsilon +a_1 ^2+a_2 ^2} \hfill \\
+\end{array} }} \right]
+\end{equation}
+where ($a_1$, $a_2$) are the isopycnal slopes in ($\textbf{i}$,
+$\textbf{j}$) directions, relative to geopotentials:
+\begin{equation*}
+a_1 =\frac{e_3 }{e_1 }\left( {\frac{\partial \rho }{\partial i}} \right)\left( {\frac{\partial \rho }{\partial k}} \right)^{-1}
+\qquad , \qquad
+a_2 =\frac{e_3 }{e_2 }\left( {\frac{\partial \rho }{\partial j}}
+\right)\left( {\frac{\partial \rho }{\partial k}} \right)^{-1}
+\end{equation*}
+
+In practice, isopycnal slopes are generally less than $10^{-2}$ in the ocean, so
+$\textbf {A}_{\textbf I}$ can be simplified appreciably \citep{Cox1987}:
+\begin{subequations} \label{Apdx_B4}
+\begin{equation} \label{Apdx_B4a}
+{\textbf{A}_{\textbf{I}}} \approx A^{lT}\;\Re\;\text{where} \;\Re =
+\left[ {{\begin{array}{*{20}c}
+ 1 \hfill & 0 \hfill & {-a_1 } \hfill \\
+ 0 \hfill & 1 \hfill & {-a_2 } \hfill \\
+ {-a_1 } \hfill & {-a_2 } \hfill & {\varepsilon +a_1 ^2+a_2 ^2} \hfill \\
+\end{array} }} \right],
+\end{equation}
+and the iso/dianeutral diffusive operator in $z$-coordinates is then
+\begin{equation}\label{Apdx_B4b}
+ D^T = \left. \nabla \right|_z \cdot
+ \left[ A^{lT} \;\Re \cdot \left. \nabla \right|_z T \right]. \\
+\end{equation}
+\end{subequations}
+
+
+Physically, the full tensor \eqref{Apdx_B3}
+represents strong isoneutral diffusion on a plane parallel to the isoneutral
+surface and weak dianeutral diffusion perpendicular to this plane.
+However, the approximate `weak-slope' tensor \eqref{Apdx_B4a} represents strong
+diffusion along the isoneutral surface, with weak
+\emph{vertical} diffusion -- the principal axes of the tensor are no
+longer orthogonal. This simplification also decouples
+the ($i$,$z$) and ($j$,$z$) planes of the tensor. The weak-slope operator therefore takes the same
+form, \eqref{Apdx_B4}, as \eqref{Apdx_B2}, the diffusion operator for geopotential
+diffusion written in non-orthogonal $i,j,s$-coordinates. Written out
+explicitly,
+
+\begin{multline} \label{Apdx_B_ldfiso}
+ D^T=\frac{1}{e_1 e_2 }\left\{
+ {\;\frac{\partial }{\partial i}\left[ {A_h \left( {\frac{e_2}{e_1}\frac{\partial T}{\partial i}-a_1 \frac{e_2}{e_3}\frac{\partial T}{\partial k}} \right)} \right]}
+ {+\frac{\partial}{\partial j}\left[ {A_h \left( {\frac{e_1}{e_2}\frac{\partial T}{\partial j}-a_2 \frac{e_1}{e_3}\frac{\partial T}{\partial k}} \right)} \right]\;} \right\} \\
+\shoveright{+\frac{1}{e_3 }\frac{\partial }{\partial k}\left[ {A_h \left( {-\frac{a_1 }{e_1 }\frac{\partial T}{\partial i}-\frac{a_2 }{e_2 }\frac{\partial T}{\partial j}+\frac{\left( {a_1 ^2+a_2 ^2+\varepsilon} \right)}{e_3 }\frac{\partial T}{\partial k}} \right)} \right]}. \\
+\end{multline}
+
+
+The isopycnal diffusion operator \eqref{Apdx_B4},
+\eqref{Apdx_B_ldfiso} conserves tracer quantity and dissipates its
+square. The demonstration of the first property is trivial as \eqref{Apdx_B4} is the divergence
+of fluxes. Let us demonstrate the second one:
+\begin{equation*}
+\iiint\limits_D T\;\nabla .\left( {\textbf{A}}_{\textbf{I}} \nabla T \right)\,dv
+ = -\iiint\limits_D \nabla T\;.\left( {\textbf{A}}_{\textbf{I}} \nabla T \right)\,dv,
+\end{equation*}
+and since
+\begin{subequations}
+\begin{align*} {\begin{array}{*{20}l}
+\nabla T\;.\left( {{\rm {\bf A}}_{\rm {\bf I}} \nabla T}
+\right)&=A^{lT}\left[ {\left( {\frac{\partial T}{\partial i}} \right)^2-2a_1
+\frac{\partial T}{\partial i}\frac{\partial T}{\partial k}+\left(
+{\frac{\partial T}{\partial j}} \right)^2} \right. \\
+&\qquad \qquad \qquad
+{ \left. -\,{2a_2 \frac{\partial T}{\partial j}\frac{\partial T}{\partial k}+\left( {a_1 ^2+a_2 ^2+\varepsilon} \right)\left( {\frac{\partial T}{\partial k}} \right)^2} \right]} \\
+&=A_h \left[ {\left( {\frac{\partial T}{\partial i}-a_1 \frac{\partial
+ T}{\partial k}} \right)^2+\left( {\frac{\partial T}{\partial
+ j}-a_2 \frac{\partial T}{\partial k}} \right)^2}
+ +\varepsilon \left(\frac{\partial T}{\partial k}\right) ^2\right] \\
+& \geq 0
+\end{array} }
+\end{align*}
+\end{subequations}
+\addtocounter{equation}{-1}
+ the property becomes obvious.
+
+\subsubsection*{In generalized vertical coordinates}
+
+Because the weak-slope operator \eqref{Apdx_B4}, \eqref{Apdx_B_ldfiso} is decoupled
+in the ($i$,$z$) and ($j$,$z$) planes, it may be transformed into
+generalized $s$-coordinates in the same way as \eqref{Apdx_B_1} was transformed into
+\eqref{Apdx_B_2}. The resulting operator then takes the simple form
+
+\begin{equation} \label{Apdx_B_ldfiso_s}
+D^T = \left. \nabla \right|_s \cdot
+ \left[ A^{lT} \;\Re \cdot \left. \nabla \right|_s T \right] \\
+\;\;\text{where} \;\Re =\left( {{\begin{array}{*{20}c}
+ 1 \hfill & 0 \hfill & {-r _1 } \hfill \\
+ 0 \hfill & 1 \hfill & {-r _2 } \hfill \\
+ {-r _1 } \hfill & {-r _2 } \hfill & {\varepsilon +r _1
+^2+r _2 ^2} \hfill \\
+\end{array} }} \right),
+\end{equation}
+
+where ($r_1$, $r_2$) are the isopycnal slopes in ($\textbf{i}$,
+$\textbf{j}$) directions, relative to $s$-coordinate surfaces:
+\begin{equation*}
+r_1 =\frac{e_3 }{e_1 }\left( {\frac{\partial \rho }{\partial i}} \right)\left( {\frac{\partial \rho }{\partial s}} \right)^{-1}
+\qquad , \qquad
+r_2 =\frac{e_3 }{e_2 }\left( {\frac{\partial \rho }{\partial j}}
+\right)\left( {\frac{\partial \rho }{\partial s}} \right)^{-1}.
+\end{equation*}
+
+To prove \eqref{Apdx_B5} by direct re-expression of \eqref{Apdx_B_ldfiso} is
+straightforward, but laborious. An easier way is first to note (by reversing the
+derivation of \eqref{Apdx_B_2} from \eqref{Apdx_B_1} ) that the
+weak-slope operator may be \emph{exactly} reexpressed in
+non-orthogonal $i,j,\rho$-coordinates as
+
+\begin{equation} \label{Apdx_B5}
+D^T = \left. \nabla \right|_\rho \cdot
+ \left[ A^{lT} \;\Re \cdot \left. \nabla \right|_\rho T \right] \\
+\;\;\text{where} \;\Re =\left( {{\begin{array}{*{20}c}
+ 1 \hfill & 0 \hfill &0 \hfill \\
+ 0 \hfill & 1 \hfill & 0 \hfill \\
+0 \hfill & 0 \hfill & \varepsilon \hfill \\
+\end{array} }} \right).
+\end{equation}
+Then direct transformation from $i,j,\rho$-coordinates to
+$i,j,s$-coordinates gives \eqref{Apdx_B_ldfiso_s} immediately.
+
+Note that the weak-slope approximation is only made in
+transforming from the (rotated,orthogonal) isoneutral axes to the
+non-orthogonal $i,j,\rho$-coordinates. The further transformation
+into $i,j,s$-coordinates is exact, whatever the steepness of
+the $s$-surfaces, in the same way as the transformation of
+horizontal/vertical Laplacian diffusion in $z$-coordinates,
+\eqref{Apdx_B_1} onto $s$-coordinates is exact, however steep the $s$-surfaces.
+
+
+% ================================================================
+% Lateral/Vertical Momentum Diffusive Operators
+% ================================================================
+\section{Lateral/Vertical Momentum Diffusive Operators}
+\label{Apdx_B_3}
+
+The second order momentum diffusion operator (Laplacian) in the $z$-coordinate
+is found by applying \eqref{Eq_PE_lap_vector}, the expression for the Laplacian
+of a vector, to the horizontal velocity vector :
+\begin{align*}
+\Delta {\textbf{U}}_h
+&=\nabla \left( {\nabla \cdot {\textbf{U}}_h } \right)-
+\nabla \times \left( {\nabla \times {\textbf{U}}_h } \right) \\
+\\
+&=\left( {{\begin{array}{*{20}c}
+ {\frac{1}{e_1 }\frac{\partial \chi }{\partial i}} \hfill \\
+ {\frac{1}{e_2 }\frac{\partial \chi }{\partial j}} \hfill \\
+ {\frac{1}{e_3 }\frac{\partial \chi }{\partial k}} \hfill \\
+\end{array} }} \right)-\left( {{\begin{array}{*{20}c}
+ {\frac{1}{e_2 }\frac{\partial \zeta }{\partial j}-\frac{1}{e_3
+}\frac{\partial }{\partial k}\left( {\frac{1}{e_3 }\frac{\partial
+u}{\partial k}} \right)} \hfill \\
+ {\frac{1}{e_3 }\frac{\partial }{\partial k}\left( {-\frac{1}{e_3
+}\frac{\partial v}{\partial k}} \right)-\frac{1}{e_1 }\frac{\partial \zeta
+}{\partial i}} \hfill \\
+ {\frac{1}{e_1 e_2 }\left[ {\frac{\partial }{\partial i}\left( {\frac{e_2
+}{e_3 }\frac{\partial u}{\partial k}} \right)-\frac{\partial }{\partial
+j}\left( {-\frac{e_1 }{e_3 }\frac{\partial v}{\partial k}} \right)} \right]}
+\hfill \\
+\end{array} }} \right)
+\\
+\\
+&=\left( {{\begin{array}{*{20}c}
+{\frac{1}{e_1 }\frac{\partial \chi }{\partial i}-\frac{1}{e_2 }\frac{\partial \zeta }{\partial j}} \\
+{\frac{1}{e_2 }\frac{\partial \chi }{\partial j}+\frac{1}{e_1 }\frac{\partial \zeta }{\partial i}} \\
+0 \\
+\end{array} }} \right)
++\frac{1}{e_3 }
+\left( {{\begin{array}{*{20}c}
+{\frac{\partial }{\partial k}\left( {\frac{1}{e_3 }\frac{\partial u}{\partial k}} \right)} \\
+{\frac{\partial }{\partial k}\left( {\frac{1}{e_3 }\frac{\partial v}{\partial k}} \right)} \\
+{\frac{\partial \chi }{\partial k}-\frac{1}{e_1 e_2 }\left( {\frac{\partial ^2\left( {e_2 \,u} \right)}{\partial i\partial k}+\frac{\partial ^2\left( {e_1 \,v} \right)}{\partial j\partial k}} \right)} \\
+\end{array} }} \right)
+\end{align*}
+Using \eqref{Eq_PE_div}, the definition of the horizontal divergence, the third
+componant of the second vector is obviously zero and thus :
+\begin{equation*}
+\Delta {\textbf{U}}_h = \nabla _h \left( \chi \right) - \nabla _h \times \left( \zeta \right) + \frac {1}{e_3 } \frac {\partial }{\partial k} \left( {\frac {1}{e_3 } \frac{\partial {\textbf{ U}}_h }{\partial k}} \right)
+\end{equation*}
+
+Note that this operator ensures a full separation between the vorticity and horizontal
+divergence fields (see Appendix~\ref{Apdx_C}). It is only equal to a Laplacian
+applied to each component in Cartesian coordinates, not on the sphere.
+
+The horizontal/vertical second order (Laplacian type) operator used to diffuse
+horizontal momentum in the $z$-coordinate therefore takes the following form :
+\begin{equation} \label{Apdx_B_Lap_U}
+ {\textbf{D}}^{\textbf{U}} =
+ \nabla _h \left( {A^{lm}\;\chi } \right)
+ - \nabla _h \times \left( {A^{lm}\;\zeta \;{\textbf{k}}} \right)
+ + \frac{1}{e_3 }\frac{\partial }{\partial k}\left( {\frac{A^{vm}\;}{e_3 }
+ \frac{\partial {\rm {\bf U}}_h }{\partial k}} \right) \\
+\end{equation}
+that is, in expanded form:
+\begin{align*}
+D^{\textbf{U}}_u
+& = \frac{1}{e_1} \frac{\partial \left( {A^{lm}\chi } \right)}{\partial i}
+ -\frac{1}{e_2} \frac{\partial \left( {A^{lm}\zeta } \right)}{\partial j}
+ +\frac{1}{e_3} \frac{\partial u}{\partial k} \\
+D^{\textbf{U}}_v
+& = \frac{1}{e_2 }\frac{\partial \left( {A^{lm}\chi } \right)}{\partial j}
+ +\frac{1}{e_1 }\frac{\partial \left( {A^{lm}\zeta } \right)}{\partial i}
+ +\frac{1}{e_3} \frac{\partial v}{\partial k}
+\end{align*}
+
+Note Bene: introducing a rotation in \eqref{Apdx_B_Lap_U} does not lead to a
+useful expression for the iso/diapycnal Laplacian operator in the $z$-coordinate.
+Similarly, we did not found an expression of practical use for the geopotential
+horizontal/vertical Laplacian operator in the $s$-coordinate. Generally,
+\eqref{Apdx_B_Lap_U} is used in both $z$- and $s$-coordinate systems, that is
+a Laplacian diffusion is applied on momentum along the coordinate directions.
Index: /branches/UKMO/dev_r5518_GO6_under_ice_relax_dr_hook/DOC/TexFiles/Chapters/Annex_C.tex
===================================================================
--- /branches/UKMO/dev_r5518_GO6_under_ice_relax_dr_hook/DOC/TexFiles/Chapters/Annex_C.tex (revision 11643)
+++ /branches/UKMO/dev_r5518_GO6_under_ice_relax_dr_hook/DOC/TexFiles/Chapters/Annex_C.tex (revision 11643)
@@ -0,0 +1,1549 @@
+% ================================================================
+% Chapter Ñ Appendix C : Discrete Invariants of the Equations
+% ================================================================
+\chapter{Discrete Invariants of the Equations}
+\label{Apdx_C}
+\minitoc
+
+%%% Appendix put in gmcomment as it has not been updated for z* and s coordinate
+%I'm writting this appendix. It will be available in a forthcoming release of the documentation
+
+%\gmcomment{
+
+\newpage
+$\ $\newline % force a new ligne
+
+% ================================================================
+% Introduction / Notations
+% ================================================================
+\section{Introduction / Notations}
+\label{Apdx_C.0}
+
+Notation used in this appendix in the demonstations :
+
+fluxes at the faces of a $T$-box:
+\begin{equation*}
+U = e_{2u}\,e_{3u}\; u \qquad V = e_{1v}\,e_{3v}\; v \qquad W = e_{1w}\,e_{2w}\; \omega \\
+\end{equation*}
+
+volume of cells at $u$-, $v$-, and $T$-points:
+\begin{equation*}
+b_u = e_{1u}\,e_{2u}\,e_{3u} \qquad b_v = e_{1v}\,e_{2v}\,e_{3v} \qquad b_t = e_{1t}\,e_{2t}\,e_{3t} \\
+\end{equation*}
+
+partial derivative notation: $\partial_\bullet = \frac{\partial}{\partial \bullet}$
+
+$dv=e_1\,e_2\,e_3 \,di\,dj\,dk$ is the volume element, with only $e_3$ that depends on time.
+$D$ and $S$ are the ocean domain volume and surface, respectively.
+No wetting/drying is allow ($i.e.$ $\frac{\partial S}{\partial t} = 0$)
+Let $k_s$ and $k_b$ be the ocean surface and bottom, resp.
+($i.e.$ $s(k_s) = \eta$ and $s(k_b)=-H$, where $H$ is the bottom depth).
+\begin{flalign*}
+ z(k) = \eta - \int\limits_{\tilde{k}=k}^{\tilde{k}=k_s} e_3(\tilde{k}) \;d\tilde{k}
+ = \eta - \int\limits_k^{k_s} e_3 \;d\tilde{k}
+\end{flalign*}
+
+Continuity equation with the above notation:
+\begin{equation*}
+\frac{1}{e_{3t}} \partial_t (e_{3t})+ \frac{1}{b_t} \biggl\{ \delta_i [U] + \delta_j [V] + \delta_k [W] \biggr\} = 0
+\end{equation*}
+
+A quantity, $Q$ is conserved when its domain averaged time change is zero, that is when:
+\begin{equation*}
+\partial_t \left( \int_D{ Q\;dv } \right) =0
+\end{equation*}
+Noting that the coordinate system used .... blah blah
+\begin{equation*}
+\partial_t \left( \int_D {Q\;dv} \right) = \int_D { \partial_t \left( e_3 \, Q \right) e_1e_2\;di\,dj\,dk }
+ = \int_D { \frac{1}{e_3} \partial_t \left( e_3 \, Q \right) dv } =0
+\end{equation*}
+equation of evolution of $Q$ written as the time evolution of the vertical content of $Q$
+like for tracers, or momentum in flux form, the quadratic quantity $\frac{1}{2}Q^2$ is conserved when :
+\begin{flalign*}
+\partial_t \left( \int_D{ \frac{1}{2} \,Q^2\;dv } \right)
+=& \int_D{ \frac{1}{2} \partial_t \left( \frac{1}{e_3}\left( e_3 \, Q \right)^2 \right) e_1e_2\;di\,dj\,dk } \\
+=& \int_D { Q \;\partial_t\left( e_3 \, Q \right) e_1e_2\;di\,dj\,dk }
+- \int_D { \frac{1}{2} Q^2 \,\partial_t (e_3) \;e_1e_2\;di\,dj\,dk } \\
+\end{flalign*}
+that is in a more compact form :
+\begin{flalign} \label{Eq_Q2_flux}
+\partial_t \left( \int_D {\frac{1}{2} Q^2\;dv} \right)
+=& \int_D { \frac{Q}{e_3} \partial_t \left( e_3 \, Q \right) dv }
+ - \frac{1}{2} \int_D { \frac{Q^2}{e_3} \partial_t (e_3) \;dv }
+\end{flalign}
+equation of evolution of $Q$ written as the time evolution of $Q$
+like for momentum in vector invariant form, the quadratic quantity $\frac{1}{2}Q^2$ is conserved when :
+\begin{flalign*}
+\partial_t \left( \int_D {\frac{1}{2} Q^2\;dv} \right)
+=& \int_D { \frac{1}{2} \partial_t \left( e_3 \, Q^2 \right) \;e_1e_2\;di\,dj\,dk } \\
+=& \int_D { Q \partial_t Q \;e_1e_2e_3\;di\,dj\,dk }
++ \int_D { \frac{1}{2} Q^2 \, \partial_t e_3 \;e_1e_2\;di\,dj\,dk } \\
+\end{flalign*}
+that is in a more compact form :
+\begin{flalign} \label{Eq_Q2_vect}
+\partial_t \left( \int_D {\frac{1}{2} Q^2\;dv} \right)
+=& \int_D { Q \,\partial_t Q \;dv }
++ \frac{1}{2} \int_D { \frac{1}{e_3} Q^2 \partial_t e_3 \;dv }
+\end{flalign}
+
+
+% ================================================================
+% Continuous Total energy Conservation
+% ================================================================
+\section{Continuous conservation}
+\label{Apdx_C.1}
+
+
+The discretization of pimitive equation in $s$-coordinate ($i.e.$ time and space varying
+vertical coordinate) must be chosen so that the discrete equation of the model satisfy
+integral constrains on energy and enstrophy.
+
+
+Let us first establish those constraint in the continuous world.
+The total energy ($i.e.$ kinetic plus potential energies) is conserved :
+\begin{flalign} \label{Eq_Tot_Energy}
+ \partial_t \left( \int_D \left( \frac{1}{2} {\textbf{U}_h}^2 + \rho \, g \, z \right) \;dv \right) = & 0
+\end{flalign}
+under the following assumptions: no dissipation, no forcing
+(wind, buoyancy flux, atmospheric pressure variations), mass
+conservation, and closed domain.
+
+This equation can be transformed to obtain several sub-equalities.
+The transformation for the advection term depends on whether
+the vector invariant form or the flux form is used for the momentum equation.
+Using \eqref{Eq_Q2_vect} and introducing \eqref{Apdx_A_dyn_vect} in \eqref{Eq_Tot_Energy}
+for the former form and
+Using \eqref{Eq_Q2_flux} and introducing \eqref{Apdx_A_dyn_flux} in \eqref{Eq_Tot_Energy}
+for the latter form leads to:
+
+\begin{subequations} \label{E_tot}
+
+advection term (vector invariant form):
+\begin{equation} \label{E_tot_vect_vor}
+\int\limits_D \zeta \; \left( \textbf{k} \times \textbf{U}_h \right) \cdot \textbf{U}_h \; dv = 0 \\
+\end{equation}
+%
+\begin{equation} \label{E_tot_vect_adv}
+ \int\limits_D \textbf{U}_h \cdot \nabla_h \left( \frac{{\textbf{U}_h}^2}{2} \right) dv
++ \int\limits_D \textbf{U}_h \cdot \nabla_z \textbf{U}_h \;dv
+- \int\limits_D { \frac{{\textbf{U}_h}^2}{2} \frac{1}{e_3} \partial_t e_3 \;dv } = 0 \\
+\end{equation}
+
+advection term (flux form):
+\begin{equation} \label{E_tot_flux_metric}
+\int\limits_D \frac{1} {e_1 e_2 } \left( v \,\partial_i e_2 - u \,\partial_j e_1 \right)\;
+ \left( \textbf{k} \times \textbf{U}_h \right) \cdot \textbf{U}_h \; dv = 0 \\
+\end{equation}
+
+\begin{equation} \label{E_tot_flux_adv}
+ \int\limits_D \textbf{U}_h \cdot \left( {{\begin{array} {*{20}c}
+\nabla \cdot \left( \textbf{U}\,u \right) \hfill \\
+\nabla \cdot \left( \textbf{U}\,v \right) \hfill \\ \end{array}} } \right) \;dv
++ \frac{1}{2} \int\limits_D { {\textbf{U}_h}^2 \frac{1}{e_3} \partial_t e_3 \;dv } =\;0 \\
+\end{equation}
+
+coriolis term
+\begin{equation} \label{E_tot_cor}
+\int\limits_D f \; \left( \textbf{k} \times \textbf{U}_h \right) \cdot \textbf{U}_h \; dv = 0 \\
+\end{equation}
+
+pressure gradient:
+\begin{equation} \label{E_tot_pg}
+ - \int\limits_D \left. \nabla p \right|_z \cdot \textbf{U}_h \;dv
+= - \int\limits_D \nabla \cdot \left( \rho \,\textbf {U} \right)\;g\;z\;\;dv
+ + \int\limits_D g\, \rho \; \partial_t z \;dv \\
+\end{equation}
+\end{subequations}
+
+where $\nabla_h = \left. \nabla \right|_k$ is the gradient along the $s$-surfaces.
+
+blah blah....
+$\ $\newline % force a new ligne
+The prognostic ocean dynamics equation can be summarized as follows:
+\begin{equation*}
+\text{NXT} = \dbinom {\text{VOR} + \text{KEG} + \text {ZAD} }
+ {\text{COR} + \text{ADV} }
+ + \text{HPG} + \text{SPG} + \text{LDF} + \text{ZDF}
+\end{equation*}
+$\ $\newline % force a new ligne
+
+Vector invariant form:
+\begin{subequations} \label{E_tot_vect}
+\begin{equation} \label{E_tot_vect_vor}
+\int\limits_D \textbf{U}_h \cdot \text{VOR} \;dv = 0 \\
+\end{equation}
+\begin{equation} \label{E_tot_vect_adv}
+ \int\limits_D \textbf{U}_h \cdot \text{KEG} \;dv
++ \int\limits_D \textbf{U}_h \cdot \text{ZAD} \;dv
+- \int\limits_D { \frac{{\textbf{U}_h}^2}{2} \frac{1}{e_3} \partial_t e_3 \;dv } = 0 \\
+\end{equation}
+\begin{equation} \label{E_tot_pg}
+ - \int\limits_D \textbf{U}_h \cdot (\text{HPG}+ \text{SPG}) \;dv
+= - \int\limits_D \nabla \cdot \left( \rho \,\textbf {U} \right)\;g\;z\;\;dv
+ + \int\limits_D g\, \rho \; \partial_t z \;dv \\
+\end{equation}
+\end{subequations}
+
+Flux form:
+\begin{subequations} \label{E_tot_flux}
+\begin{equation} \label{E_tot_flux_metric}
+\int\limits_D \textbf{U}_h \cdot \text {COR} \; dv = 0 \\
+\end{equation}
+\begin{equation} \label{E_tot_flux_adv}
+ \int\limits_D \textbf{U}_h \cdot \text{ADV} \;dv
++ \frac{1}{2} \int\limits_D { {\textbf{U}_h}^2 \frac{1}{e_3} \partial_t e_3 \;dv } =\;0 \\
+\end{equation}
+\begin{equation} \label{E_tot_pg}
+ - \int\limits_D \textbf{U}_h \cdot (\text{HPG}+ \text{SPG}) \;dv
+= - \int\limits_D \nabla \cdot \left( \rho \,\textbf {U} \right)\;g\;z\;\;dv
+ + \int\limits_D g\, \rho \; \partial_t z \;dv \\
+\end{equation}
+\end{subequations}
+
+
+$\ $\newline % force a new ligne
+
+
+\eqref{E_tot_pg} is the balance between the conversion KE to PE and PE to KE.
+Indeed the left hand side of \eqref{E_tot_pg} can be transformed as follows:
+\begin{flalign*}
+\partial_t \left( \int\limits_D { \rho \, g \, z \;dv} \right)
+&= + \int\limits_D \frac{1}{e_3} \partial_t (e_3\,\rho) \;g\;z\;\;dv
+ + \int\limits_D g\, \rho \; \partial_t z \;dv &&&\\
+&= - \int\limits_D \nabla \cdot \left( \rho \,\textbf {U} \right)\;g\;z\;\;dv
+ + \int\limits_D g\, \rho \; \partial_t z \;dv &&&\\
+&= + \int\limits_D \rho \,g \left( \textbf {U}_h \cdot \nabla_h z + \omega \frac{1}{e_3} \partial_k z \right) \;dv
+ + \int\limits_D g\, \rho \; \partial_t z \;dv &&&\\
+&= + \int\limits_D \rho \,g \left( \omega + \partial_t z + \textbf {U}_h \cdot \nabla_h z \right) \;dv &&&\\
+&=+ \int\limits_D g\, \rho \; w \; dv &&&\\
+\end{flalign*}
+where the last equality is obtained by noting that the brackets is exactly the expression of $w$,
+the vertical velocity referenced to the fixe $z$-coordinate system (see \eqref{Apdx_A_w_s}).
+
+The left hand side of \eqref{E_tot_pg} can be transformed as follows:
+\begin{flalign*}
+- \int\limits_D \left. \nabla p \right|_z & \cdot \textbf{U}_h \;dv
+= - \int\limits_D \left( \nabla_h p + \rho \, g \nabla_h z \right) \cdot \textbf{U}_h \;dv &&&\\
+\allowdisplaybreaks
+&= - \int\limits_D \nabla_h p \cdot \textbf{U}_h \;dv - \int\limits_D \rho \, g \, \nabla_h z \cdot \textbf{U}_h \;dv &&&\\
+\allowdisplaybreaks
+&= +\int\limits_D p \,\nabla_h \cdot \textbf{U}_h \;dv + \int\limits_D \rho \, g \left( \omega - w + \partial_t z \right) \;dv &&&\\
+\allowdisplaybreaks
+&= -\int\limits_D p \left( \frac{1}{e_3} \partial_t e_3 + \frac{1}{e_3} \partial_k \omega \right) \;dv
+ +\int\limits_D \rho \, g \left( \omega - w + \partial_t z \right) \;dv &&&\\
+\allowdisplaybreaks
+&= -\int\limits_D \frac{p}{e_3} \partial_t e_3 \;dv
+ +\int\limits_D \frac{1}{e_3} \partial_k p\; \omega \;dv
+ +\int\limits_D \rho \, g \left( \omega - w + \partial_t z \right) \;dv &&&\\
+&= -\int\limits_D \frac{p}{e_3} \partial_t e_3 \;dv
+ -\int\limits_D \rho \, g \, \omega \;dv
+ +\int\limits_D \rho \, g \left( \omega - w + \partial_t z \right) \;dv &&&\\
+&= - \int\limits_D \frac{p}{e_3} \partial_t e_3 \; \;dv
+ - \int\limits_D \rho \, g \, w \;dv
+ + \int\limits_D \rho \, g \, \partial_t z \;dv &&&\\
+\allowdisplaybreaks
+\intertext{introducing the hydrostatic balance $\partial_k p=-\rho \,g\,e_3$ in the last term,
+it becomes:}
+&= - \int\limits_D \frac{p}{e_3} \partial_t e_3 \;dv
+ - \int\limits_D \rho \, g \, w \;dv
+ - \int\limits_D \frac{1}{e_3} \partial_k p\, \partial_t z \;dv &&&\\
+&= - \int\limits_D \frac{p}{e_3} \partial_t e_3 \;dv
+ - \int\limits_D \rho \, g \, w \;dv
+ + \int\limits_D \,\frac{p}{e_3}\partial_t ( \partial_k z ) dv &&&\\
+%
+&= - \int\limits_D \rho \, g \, w \;dv &&&\\
+\end{flalign*}
+
+
+%gm comment
+\gmcomment{
+%
+The last equality comes from the following equation,
+\begin{flalign*}
+\int\limits_D p \frac{1}{e_3} \frac{\partial e_3}{\partial t}\; \;dv
+ = \int\limits_D \rho \, g \, \frac{\partial z }{\partial t} \;dv \quad, \\
+\end{flalign*}
+that can be demonstrated as follows:
+
+\begin{flalign*}
+\int\limits_D \rho \, g \, \frac{\partial z }{\partial t} \;dv
+&= \int\limits_D \rho \, g \, \frac{\partial \eta}{\partial t} \;dv
+ - \int\limits_D \rho \, g \, \frac{\partial}{\partial t} \left( \int\limits_k^{k_s} e_3 \;d\tilde{k} \right) \;dv &&&\\
+&= \int\limits_D \rho \, g \, \frac{\partial \eta}{\partial t} \;dv
+ - \int\limits_D \rho \, g \left( \int\limits_k^{k_s} \frac{\partial e_3}{\partial t} \;d\tilde{k} \right) \;dv &&&\\
+%
+\allowdisplaybreaks
+\intertext{The second term of the right hand side can be transformed by applying the integration by part rule:
+$\left[ a\,b \right]_{k_b}^{k_s} = \int_{k_b}^{k_s} a\,\frac{\partial b}{\partial k} \;dk
+ + \int_{k_b}^{k_s} \frac{\partial a}{\partial k} \,b \;dk $
+to the following function: $a= \int_k^{k_s} \frac{\partial e_3}{\partial t} \;d\tilde{k}$
+and $b= \int_k^{k_s} \rho \, e_3 \;d\tilde{k}$
+(note that $\frac{\partial}{\partial k} \left( \int_k^{k_s} a \;d\tilde{k} \right) = - a$ as $k$ is the lower bound of the integral).
+This leads to: }
+\end{flalign*}
+\begin{flalign*}
+&\left[ \int\limits_{k}^{k_s} \frac{\partial e_3}{\partial t} \,dk \cdot \int\limits_{k}^{k_s} \rho \, e_3 \,dk \right]_{k_b}^{k_s}
+=-\int\limits_{k_b}^{k_s} \left( \int\limits_k^{k_s} \frac{\partial e_3}{\partial t} \;d\tilde{k} \right) \rho \,e_3 \;dk
+ -\int\limits_{k_b}^{k_s} \frac{\partial e_3}{\partial t} \left( \int\limits_k^{k_s} \rho \, e_3 \;d\tilde{k} \right) dk
+&&&\\
+\allowdisplaybreaks
+\intertext{Noting that $\frac{\partial \eta}{\partial t}
+ = \frac{\partial}{\partial t} \left( \int_{k_b}^{k_s} e_3 \;d\tilde{k} \right)
+ = \int_{k_b}^{k_s} \frac{\partial e_3}{\partial t} \;d\tilde{k}$
+and
+ $p(k) = \int_k^{k_s} \rho \,g \, e_3 \;d\tilde{k} $,
+but also that $\frac{\partial \eta}{\partial t}$ does not depends on $k$, it comes:
+}
+& - \int\limits_{k_b}^{k_s} \rho \, \frac{\partial \eta}{\partial t} \, e_3 \;dk
+= - \int\limits_{k_b}^{k_s} \left( \int\limits_k^{k_s} \frac{\partial e_3}{\partial t} \;d\tilde{k} \right) \, \rho \, g e_3\;dk
+ - \int\limits_{k_b}^{k_s} \frac{\partial e_3}{\partial t} \frac{p}{g} \;dk &&&\\
+\end{flalign*}
+Mutliplying by $g$ and integrating over the $(i,j)$ domain it becomes:
+\begin{flalign*}
+\int\limits_D \rho \, g \, \left( \int\limits_k^{k_s} \frac{\partial e_3}{\partial t} \;d\tilde{k} \right) \;dv
+= \int\limits_D \rho \, g \, \frac{\partial \eta}{\partial t} dv
+ - \int\limits_D \frac{p}{e_3}\frac{\partial e_3}{\partial t} \;dv
+\end{flalign*}
+Using this property, we therefore have:
+\begin{flalign*}
+\int\limits_D \rho \, g \, \frac{\partial z }{\partial t} \;dv
+&= \int\limits_D \rho \, g \, \frac{\partial \eta}{\partial t} \;dv
+ - \left( \int\limits_D \rho \, g \, \frac{\partial \eta}{\partial t} dv
+ - \int\limits_D \frac{p}{e_3}\frac{\partial e_3}{\partial t} \;dv \right) &&&\\
+%
+&=\int\limits_D \frac{p}{e_3} \frac{\partial (e_3\,\rho)}{\partial t}\; \;dv
+\end{flalign*}
+% end gm comment
+}
+%
+
+
+% ================================================================
+% Discrete Total energy Conservation : vector invariant form
+% ================================================================
+\section{Discrete total energy conservation : vector invariant form}
+\label{Apdx_C.1}
+
+% -------------------------------------------------------------------------------------------------------------
+% Total energy conservation
+% -------------------------------------------------------------------------------------------------------------
+\subsection{Total energy conservation}
+\label{Apdx_C_KE+PE}
+
+The discrete form of the total energy conservation, \eqref{Eq_Tot_Energy}, is given by:
+\begin{flalign*}
+\partial_t \left( \sum\limits_{i,j,k} \biggl\{ \frac{u^2}{2} \,b_u + \frac{v^2}{2}\, b_v + \rho \, g \, z_t \,b_t \biggr\} \right) &=0 \\
+\end{flalign*}
+which in vector invariant forms, it leads to:
+\begin{equation} \label{KE+PE_vect_discrete} \begin{split}
+ \sum\limits_{i,j,k} \biggl\{ u\, \partial_t u \;b_u
+ + v\, \partial_t v \;b_v \biggr\}
+ + \frac{1}{2} \sum\limits_{i,j,k} \biggl\{ \frac{u^2}{e_{3u}}\partial_t e_{3u} \;b_u
+ + \frac{v^2}{e_{3v}}\partial_t e_{3v} \;b_v \biggr\} \\
+= - \sum\limits_{i,j,k} \biggl\{ \frac{1}{e_{3t}}\partial_t (e_{3t} \rho) \, g \, z_t \;b_t \biggr\}
+ - \sum\limits_{i,j,k} \biggl\{ \rho \,g\,\partial_t (z_t) \,b_t \biggr\}
+\end{split} \end{equation}
+
+Substituting the discrete expression of the time derivative of the velocity either in vector invariant,
+leads to the discrete equivalent of the four equations \eqref{E_tot_flux}.
+
+% -------------------------------------------------------------------------------------------------------------
+% Vorticity term (coriolis + vorticity part of the advection)
+% -------------------------------------------------------------------------------------------------------------
+\subsection{Vorticity term (coriolis + vorticity part of the advection)}
+\label{Apdx_C_vor}
+
+Let $q$, located at $f$-points, be either the relative ($q=\zeta / e_{3f}$), or
+the planetary ($q=f/e_{3f}$), or the total potential vorticity ($q=(\zeta +f) /e_{3f}$).
+Two discretisation of the vorticity term (ENE and EEN) allows the conservation of
+the kinetic energy.
+% -------------------------------------------------------------------------------------------------------------
+% Vorticity Term with ENE scheme
+% -------------------------------------------------------------------------------------------------------------
+\subsubsection{Vorticity Term with ENE scheme (\np{ln\_dynvor\_ene}=.true.)}
+\label{Apdx_C_vorENE}
+
+For the ENE scheme, the two components of the vorticity term are given by :
+\begin{equation*}
+- e_3 \, q \;{\textbf{k}}\times {\textbf {U}}_h \equiv
+ \left( {{ \begin{array} {*{20}c}
+ + \frac{1} {e_{1u}} \;
+ \overline {\, q \ \overline {\left( e_{1v}\,e_{3v}\,v \right)}^{\,i+1/2}} ^{\,j} \hfill \\
+ - \frac{1} {e_{2v}} \;
+ \overline {\, q \ \overline {\left( e_{2u}\,e_{3u}\,u \right)}^{\,j+1/2}} ^{\,i} \hfill \\
+ \end{array}} } \right)
+\end{equation*}
+
+This formulation does not conserve the enstrophy but it does conserve the
+total kinetic energy. Indeed, the kinetic energy tendency associated to the
+vorticity term and averaged over the ocean domain can be transformed as
+follows:
+\begin{flalign*}
+&\int\limits_D - \left( e_3 \, q \;\textbf{k} \times \textbf{U}_h \right) \cdot \textbf{U}_h \; dv && \\
+& \qquad \qquad {\begin{array}{*{20}l}
+&\equiv \sum\limits_{i,j,k} \biggl\{
+ \frac{1} {e_{1u}} \overline { \,q\ \overline{ V }^{\,i+1/2}} ^{\,j} \, u \; b_u
+ - \frac{1} {e_{2v}}\overline { \, q\ \overline{ U }^{\,j+1/2}} ^{\,i} \, v \; b_v \; \biggr\} \\
+&\equiv \sum\limits_{i,j,k} \biggl\{
+ \overline { \,q\ \overline{ V }^{\,i+1/2}}^{\,j} \; U
+ - \overline { \,q\ \overline{ U }^{\,j+1/2}}^{\,i} \; V \; \biggr\} \\
+&\equiv \sum\limits_{i,j,k} q \ \biggl\{ \overline{ V }^{\,i+1/2}\; \overline{ U }^{\,j+1/2}
+ - \overline{ U }^{\,j+1/2}\; \overline{ V }^{\,i+1/2} \biggr\} \quad \equiv 0
+\end{array} }
+\end{flalign*}
+In other words, the domain averaged kinetic energy does not change due to the vorticity term.
+
+
+% -------------------------------------------------------------------------------------------------------------
+% Vorticity Term with EEN scheme
+% -------------------------------------------------------------------------------------------------------------
+\subsubsection{Vorticity Term with EEN scheme (\np{ln\_dynvor\_een}=.true.)}
+\label{Apdx_C_vorEEN}
+
+With the EEN scheme, the vorticity terms are represented as:
+\begin{equation} \label{Eq_dynvor_een}
+\left\{ { \begin{aligned}
+ +q\,e_3 \, v &\equiv +\frac{1}{e_{1u} } \sum_{\substack{i_p,\,k_p}}
+ {^{i+1/2-i_p}_j} \mathbb{Q}^{i_p}_{j_p} \left( e_{1v} e_{3v} \ v \right)^{i+i_p-1/2}_{j+j_p} \\
+ - q\,e_3 \, u &\equiv -\frac{1}{e_{2v} } \sum_{\substack{i_p,\,k_p}}
+ {^i_{j+1/2-j_p}} \mathbb{Q}^{i_p}_{j_p} \left( e_{2u} e_{3u} \ u \right)^{i+i_p}_{j+j_p-1/2} \\
+\end{aligned} } \right.
+\end{equation}
+where the indices $i_p$ and $k_p$ take the following value:
+$i_p = -1/2$ or $1/2$ and $j_p = -1/2$ or $1/2$,
+and the vorticity triads, ${^i_j}\mathbb{Q}^{i_p}_{j_p}$, defined at $T$-point, are given by:
+\begin{equation} \label{Q_triads}
+_i^j \mathbb{Q}^{i_p}_{j_p}
+= \frac{1}{12} \ \left( q^{i-i_p}_{j+j_p} + q^{i+j_p}_{j+i_p} + q^{i+i_p}_{j-j_p} \right)
+\end{equation}
+
+This formulation does conserve the total kinetic energy. Indeed,
+\begin{flalign*}
+&\int\limits_D - \textbf{U}_h \cdot \left( \zeta \;\textbf{k} \times \textbf{U}_h \right) \; dv && \\
+\equiv \sum\limits_{i,j,k} & \biggl\{
+ \left[ \sum_{\substack{i_p,\,k_p}}
+ {^{i+1/2-i_p}_j}\mathbb{Q}^{i_p}_{j_p} \; V^{i+1/2-i_p}_{j+j_p} \right] U^{i+1/2}_{j} % &&\\
+ - \left[ \sum_{\substack{i_p,\,k_p}}
+ {^i_{j+1/2-j_p}}\mathbb{Q}^{i_p}_{j_p} \; U^{i+i_p}_{j+1/2-j_p} \right] V^{i}_{j+1/2} \biggr\} && \\
+\\
+\equiv \sum\limits_{i,j,k} & \sum_{\substack{i_p,\,k_p}} \biggl\{ \ \
+ {^{i+1/2-i_p}_j}\mathbb{Q}^{i_p}_{j_p} \; V^{i+1/2-i_p}_{j+j_p} \, U^{i+1/2}_{j} % &&\\
+ - {^i_{j+1/2-j_p}}\mathbb{Q}^{i_p}_{j_p} \; U^{i+i_p}_{j+1/2-j_p} \, V^{i}_{j+1/2} \ \; \biggr\} && \\
+%
+\allowdisplaybreaks
+\intertext{ Expending the summation on $i_p$ and $k_p$, it becomes:}
+%
+\equiv \sum\limits_{i,j,k} & \biggl\{ \ \
+ {^{i+1}_j }\mathbb{Q}^{-1/2}_{+1/2} \;V^{i+1}_{j+1/2} \; U^{\,i+1/2}_{j}
+ - {^i_{j}\quad}\mathbb{Q}^{-1/2}_{+1/2} \; U^{i-1/2}_{j} \; V^{\,i}_{j+1/2} && \\
+ & + {^{i+1}_j }\mathbb{Q}^{-1/2}_{-1/2} \; V^{i+1}_{j-1/2} \; U^{\,i+1/2}_{j}
+ - {^i_{j+1} }\mathbb{Q}^{-1/2}_{-1/2} \; U^{i-1/2}_{j+1} \; V^{\,i}_{j+1/2} \biggr. && \\
+ & + {^{i}_j\quad}\mathbb{Q}^{+1/2}_{+1/2} \; V^{i}_{j+1/2} \; U^{\,i+1/2}_{j}
+ - {^i_{j}\quad}\mathbb{Q}^{+1/2}_{+1/2} \; U^{i+1/2}_{j} \; V^{\,i}_{j+1/2} \biggr. && \\
+ & + {^{i}_j\quad}\mathbb{Q}^{+1/2}_{-1/2} \; V^{i}_{j-1/2} \; U^{\,i+1/2}_{j}
+ - {^i_{j+1} }\mathbb{Q}^{+1/2}_{-1/2} \; U^{i+1/2}_{j+1}\; V^{\,i}_{j+1/2} \ \; \biggr\} && \\
+%
+\allowdisplaybreaks
+\intertext{The summation is done over all $i$ and $j$ indices, it is therefore possible to introduce
+a shift of $-1$ either in $i$ or $j$ direction in some of the term of the summation (first term of the
+first and second lines, second term of the second and fourth lines). By doning so, we can regroup
+all the terms of the summation by triad at a ($i$,$j$) point. In other words, we regroup all the terms
+in the neighbourhood that contain a triad at the same ($i$,$j$) indices. It becomes: }
+\allowdisplaybreaks
+%
+\equiv \sum\limits_{i,j,k} & \biggl\{ \ \
+ {^{i}_j}\mathbb{Q}^{-1/2}_{+1/2} \left[ V^{i}_{j+1/2}\, U^{\,i-1/2}_{j}
+ - U^{i-1/2}_{j} \, V^{\,i}_{j+1/2} \right] && \\
+ & + {^{i}_j}\mathbb{Q}^{-1/2}_{-1/2} \left[ V^{i}_{j-1/2} \, U^{\,i-1/2}_{j}
+ - U^{i-1/2}_{j} \, V^{\,i}_{j-1/2} \right] \biggr. && \\
+ & + {^{i}_j}\mathbb{Q}^{+1/2}_{+1/2} \left[ V^{i}_{j+1/2} \, U^{\,i+1/2}_{j}
+ - U^{i+1/2}_{j} \, V^{\,i}_{j+1/2} \right] \biggr. && \\
+ & + {^{i}_j}\mathbb{Q}^{+1/2}_{-1/2} \left[ V^{i}_{j-1/2} \, U^{\,i+1/2}_{j}
+ - U^{i+1/2}_{j-1} \, V^{\,i}_{j-1/2} \right] \ \; \biggr\} \qquad
+\equiv \ 0 &&
+\end{flalign*}
+
+
+% -------------------------------------------------------------------------------------------------------------
+% Gradient of Kinetic Energy / Vertical Advection
+% -------------------------------------------------------------------------------------------------------------
+\subsubsection{Gradient of Kinetic Energy / Vertical Advection}
+\label{Apdx_C_zad}
+
+The change of Kinetic Energy (KE) due to the vertical advection is exactly
+balanced by the change of KE due to the horizontal gradient of KE~:
+\begin{equation*}
+ \int_D \textbf{U}_h \cdot \frac{1}{e_3 } \omega \partial_k \textbf{U}_h \;dv
+= - \int_D \textbf{U}_h \cdot \nabla_h \left( \frac{1}{2}\;{\textbf{U}_h}^2 \right)\;dv
+ + \frac{1}{2} \int_D { \frac{{\textbf{U}_h}^2}{e_3} \partial_t ( e_3) \;dv } \\
+\end{equation*}
+Indeed, using successively \eqref{DOM_di_adj} ($i.e.$ the skew symmetry
+property of the $\delta$ operator) and the continuity equation, then
+\eqref{DOM_di_adj} again, then the commutativity of operators
+$\overline {\,\cdot \,}$ and $\delta$, and finally \eqref{DOM_mi_adj}
+($i.e.$ the symmetry property of the $\overline {\,\cdot \,}$ operator)
+applied in the horizontal and vertical directions, it becomes:
+\begin{flalign*}
+& - \int_D \textbf{U}_h \cdot \text{KEG}\;dv
+= - \int_D \textbf{U}_h \cdot \nabla_h \left( \frac{1}{2}\;{\textbf{U}_h}^2 \right)\;dv &&&\\
+%
+\equiv & - \sum\limits_{i,j,k} \frac{1}{2} \biggl\{
+ \frac{1} {e_{1u}} \delta_{i+1/2} \left[ \overline {u^2}^{\,i} + \overline {v^2}^{\,j} \right] u \ b_u
+ + \frac{1} {e_{2v}} \delta_{j+1/2} \left[ \overline {u^2}^{\,i} + \overline {v^2}^{\,j} \right] v \ b_v \biggr\} &&& \\
+%
+\equiv & + \sum\limits_{i,j,k} \frac{1}{2} \left( \overline {u^2}^{\,i} + \overline {v^2}^{\,j} \right)\;
+ \biggl\{ \delta_{i} \left[ U \right] + \delta_{j} \left[ V \right] \biggr\} &&& \\
+\allowdisplaybreaks
+%
+\equiv & - \sum\limits_{i,j,k} \frac{1}{2}
+ \left( \overline {u^2}^{\,i} + \overline {v^2}^{\,j} \right) \;
+ \biggl\{ \frac{b_t}{e_{3t}} \partial_t (e_{3t}) + \delta_k \left[ W \right] \biggr\} &&&\\
+\allowdisplaybreaks
+%
+\equiv & + \sum\limits_{i,j,k} \frac{1}{2} \delta_{k+1/2} \left[ \overline{ u^2}^{\,i} + \overline{ v^2}^{\,j} \right] \; W
+ - \sum\limits_{i,j,k} \frac{1}{2} \left( \overline {u^2}^{\,i} + \overline {v^2}^{\,j} \right) \;\partial_t b_t &&& \\
+\allowdisplaybreaks
+%
+\equiv & + \sum\limits_{i,j,k} \frac{1} {2} \left( \overline{\delta_{k+1/2} \left[ u^2 \right]}^{\,i}
+ + \overline{\delta_{k+1/2} \left[ v^2 \right]}^{\,j} \right) \; W
+ - \sum\limits_{i,j,k} \left( \frac{u^2}{2}\,\partial_t \overline{b_t}^{\,{i+1/2}}
+ + \frac{v^2}{2}\,\partial_t \overline{b_t}^{\,{j+1/2}} \right) &&& \\
+\allowdisplaybreaks
+\intertext{Assuming that $b_u= \overline{b_t}^{\,i+1/2}$ and $b_v= \overline{b_t}^{\,j+1/2}$, or at least that the time
+derivative of these two equations is satisfied, it becomes:}
+%
+\equiv & \sum\limits_{i,j,k} \frac{1} {2}
+ \biggl\{ \; \overline{W}^{\,i+1/2}\;\delta_{k+1/2} \left[ u^2 \right]
+ + \overline{W}^{\,j+1/2}\;\delta_{k+1/2} \left[ v^2 \right] \; \biggr\}
+ - \sum\limits_{i,j,k} \left( \frac{u^2}{2}\,\partial_t b_u
+ + \frac{v^2}{2}\,\partial_t b_v \right) &&& \\
+\allowdisplaybreaks
+%
+\equiv & \sum\limits_{i,j,k}
+ \biggl\{ \; \overline{W}^{\,i+1/2}\; \overline {u}^{\,k+1/2}\; \delta_{k+1/2}[ u ]
+ + \overline{W}^{\,j+1/2}\; \overline {v}^{\,k+1/2}\; \delta_{k+1/2}[ v ] \; \biggr\}
+ - \sum\limits_{i,j,k} \left( \frac{u^2}{2}\,\partial_t b_u
+ + \frac{v^2}{2}\,\partial_t b_v \right) &&& \\
+%
+\allowdisplaybreaks
+\equiv & \sum\limits_{i,j,k}
+ \biggl\{ \; \frac{1} {b_u } \; \overline { \overline{W}^{\,i+1/2}\,\delta_{k+1/2} \left[ u \right] }^{\,k} \;u\;b_u
+ + \frac{1} {b_v } \; \overline { \overline{W}^{\,j+1/2} \delta_{k+1/2} \left[ v \right] }^{\,k} \;v\;b_v \; \biggr\}
+ - \sum\limits_{i,j,k} \left( \frac{u^2}{2}\,\partial_t b_u
+ + \frac{v^2}{2}\,\partial_t b_v \right) &&& \\
+%
+\intertext{The first term provides the discrete expression for the vertical advection of momentum (ZAD),
+while the second term corresponds exactly to \eqref{KE+PE_vect_discrete}, therefore:}
+\equiv& \int\limits_D \textbf{U}_h \cdot \text{ZAD} \;dv
+ + \frac{1}{2} \int_D { {\textbf{U}_h}^2 \frac{1}{e_3} \partial_t (e_3) \;dv } &&&\\
+\equiv& \int\limits_D \textbf{U}_h \cdot w \partial_k \textbf{U}_h \;dv
+ + \frac{1}{2} \int_D { {\textbf{U}_h}^2 \frac{1}{e_3} \partial_t (e_3) \;dv } &&&\\
+\end{flalign*}
+
+There is two main points here. First, the satisfaction of this property links the choice of
+the discrete formulation of the vertical advection and of the horizontal gradient
+of KE. Choosing one imposes the other. For example KE can also be discretized
+as $1/2\,({\overline u^{\,i}}^2 + {\overline v^{\,j}}^2)$. This leads to the following
+expression for the vertical advection:
+\begin{equation*}
+\frac{1} {e_3 }\; \omega\; \partial_k \textbf{U}_h
+\equiv \left( {{\begin{array} {*{20}c}
+\frac{1} {e_{1u}\,e_{2u}\,e_{3u}} \; \overline{\overline {e_{1t}\,e_{2t} \,\omega\;\delta_{k+1/2}
+\left[ \overline u^{\,i+1/2} \right]}}^{\,i+1/2,k} \hfill \\
+\frac{1} {e_{1v}\,e_{2v}\,e_{3v}} \; \overline{\overline {e_{1t}\,e_{2t} \,\omega \;\delta_{k+1/2}
+\left[ \overline v^{\,j+1/2} \right]}}^{\,j+1/2,k} \hfill \\
+\end{array}} } \right)
+\end{equation*}
+a formulation that requires an additional horizontal mean in contrast with
+the one used in NEMO. Nine velocity points have to be used instead of 3.
+This is the reason why it has not been chosen.
+
+Second, as soon as the chosen $s$-coordinate depends on time, an extra constraint
+arises on the time derivative of the volume at $u$- and $v$-points:
+\begin{flalign*}
+e_{1u}\,e_{2u}\,\partial_t (e_{3u}) =\overline{ e_{1t}\,e_{2t}\;\partial_t (e_{3t}) }^{\,i+1/2} \\
+e_{1v}\,e_{2v}\,\partial_t (e_{3v}) =\overline{ e_{1t}\,e_{2t}\;\partial_t (e_{3t}) }^{\,j+1/2}
+\end{flalign*}
+which is (over-)satified by defining the vertical scale factor as follows:
+\begin{flalign} \label{e3u-e3v}
+e_{3u} = \frac{1}{e_{1u}\,e_{2u}}\;\overline{ e_{1t}^{ }\,e_{2t}^{ }\,e_{3t}^{ } }^{\,i+1/2} \\
+e_{3v} = \frac{1}{e_{1v}\,e_{2v}}\;\overline{ e_{1t}^{ }\,e_{2t}^{ }\,e_{3t}^{ } }^{\,j+1/2}
+\end{flalign}
+
+Blah blah required on the the step representation of bottom topography.....
+
+
+% -------------------------------------------------------------------------------------------------------------
+% Pressure Gradient Term
+% -------------------------------------------------------------------------------------------------------------
+\subsection{Pressure Gradient Term}
+\label{Apdx_C.1.4}
+
+\gmcomment{
+A pressure gradient has no contribution to the evolution of the vorticity as the
+curl of a gradient is zero. In the $z$-coordinate, this property is satisfied locally
+on a C-grid with 2nd order finite differences (property \eqref{Eq_DOM_curl_grad}).
+}
+
+When the equation of state is linear ($i.e.$ when an advection-diffusion equation
+for density can be derived from those of temperature and salinity) the change of
+KE due to the work of pressure forces is balanced by the change of potential
+energy due to buoyancy forces:
+\begin{equation*}
+- \int_D \left. \nabla p \right|_z \cdot \textbf{U}_h \;dv
+= - \int_D \nabla \cdot \left( \rho \,\textbf {U} \right) \,g\,z \;dv
+ + \int_D g\, \rho \; \partial_t (z) \;dv
+\end{equation*}
+
+This property can be satisfied in a discrete sense for both $z$- and $s$-coordinates.
+Indeed, defining the depth of a $T$-point, $z_t$, as the sum of the vertical scale
+factors at $w$-points starting from the surface, the work of pressure forces can be
+written as:
+\begin{flalign*}
+&- \int_D \left. \nabla p \right|_z \cdot \textbf{U}_h \;dv
+\equiv \sum\limits_{i,j,k} \biggl\{ \; - \frac{1} {e_{1u}} \Bigl(
+\delta_{i+1/2} [p_t] - g\;\overline \rho^{\,i+1/2}\;\delta_{i+1/2} [z_t] \Bigr) \; u\;b_u
+ && \\ & \qquad \qquad \qquad \qquad \qquad \quad \ \,
+ - \frac{1} {e_{2v}} \Bigl(
+\delta_{j+1/2} [p_t] - g\;\overline \rho^{\,j+1/2}\delta_{j+1/2} [z_t] \Bigr) \; v\;b_v \; \biggr\} && \\
+%
+\allowdisplaybreaks
+\intertext{Using successively \eqref{DOM_di_adj}, $i.e.$ the skew symmetry property of
+the $\delta$ operator, \eqref{Eq_wzv}, the continuity equation, \eqref{Eq_dynhpg_sco},
+the hydrostatic equation in the $s$-coordinate, and $\delta_{k+1/2} \left[ z_t \right] \equiv e_{3w} $,
+which comes from the definition of $z_t$, it becomes: }
+\allowdisplaybreaks
+%
+\equiv& + \sum\limits_{i,j,k} g \biggl\{
+ \overline\rho^{\,i+1/2}\,U\,\delta_{i+1/2}[z_t]
+ + \overline\rho^{\,j+1/2}\,V\,\delta_{j+1/2}[z_t]
+ +\Bigl( \delta_i[U] + \delta_j [V] \Bigr)\;\frac{p_t}{g} \biggr\} &&\\
+%
+\equiv& + \sum\limits_{i,j,k} g \biggl\{
+ \overline\rho^{\,i+1/2}\,U\,\delta_{i+1/2}[z_t]
+ + \overline\rho^{\,j+1/2}\,V\,\delta_{j+1/2}[z_t]
+ - \left( \frac{b_t}{e_{3t}} \partial_t (e_{3t}) + \delta_k \left[ W \right] \right) \frac{p_t}{g} \biggr\} &&&\\
+%
+\equiv& + \sum\limits_{i,j,k} g \biggl\{
+ \overline\rho^{\,i+1/2}\,U\,\delta_{i+1/2}[z_t]
+ + \overline\rho^{\,j+1/2}\,V\,\delta_{j+1/2}[z_t]
+ + \frac{W}{g}\;\delta_{k+1/2} [p_t]
+ - \frac{p_t}{g}\,\partial_t b_t \biggr\} &&&\\
+%
+\equiv& + \sum\limits_{i,j,k} g \biggl\{
+ \overline\rho^{\,i+1/2}\,U\,\delta_{i+1/2}[z_t]
+ + \overline\rho^{\,j+1/2}\,V\,\delta_{j+1/2}[z_t]
+ - W\;e_{3w} \overline \rho^{\,k+1/2}
+ - \frac{p_t}{g}\,\partial_t b_t \biggr\} &&&\\
+%
+\equiv& + \sum\limits_{i,j,k} g \biggl\{
+ \overline\rho^{\,i+1/2}\,U\,\delta_{i+1/2}[z_t]
+ + \overline\rho^{\,j+1/2}\,V\,\delta_{j+1/2}[z_t]
+ + W\; \overline \rho^{\,k+1/2}\;\delta_{k+1/2} [z_t]
+ - \frac{p_t}{g}\,\partial_t b_t \biggr\} &&&\\
+%
+\allowdisplaybreaks
+%
+\equiv& - \sum\limits_{i,j,k} g \; z_t \biggl\{
+ \delta_i \left[ U\; \overline \rho^{\,i+1/2} \right]
+ + \delta_j \left[ V\; \overline \rho^{\,j+1/2} \right]
+ + \delta_k \left[ W\; \overline \rho^{\,k+1/2} \right] \biggr\}
+ - \sum\limits_{i,j,k} \biggl\{ p_t\;\partial_t b_t \biggr\} &&&\\
+%
+\equiv& + \sum\limits_{i,j,k} g \; z_t \biggl\{ \partial_t ( e_{3t} \,\rho) \biggr\} \; b_t
+ - \sum\limits_{i,j,k} \biggl\{ p_t\;\partial_t b_t \biggr\} &&&\\
+%
+\end{flalign*}
+The first term is exactly the first term of the right-hand-side of \eqref{KE+PE_vect_discrete}.
+It remains to demonstrate that the last term, which is obviously a discrete analogue of
+$\int_D \frac{p}{e_3} \partial_t (e_3)\;dv$ is equal to the last term of \eqref{KE+PE_vect_discrete}.
+In other words, the following property must be satisfied:
+\begin{flalign*}
+ \sum\limits_{i,j,k} \biggl\{ p_t\;\partial_t b_t \biggr\}
+\equiv \sum\limits_{i,j,k} \biggl\{ \rho \,g\,\partial_t (z_t) \,b_t \biggr\}
+\end{flalign*}
+
+Let introduce $p_w$ the pressure at $w$-point such that $\delta_k [p_w] = - \rho \,g\,e_{3t}$.
+The right-hand-side of the above equation can be transformed as follows:
+
+\begin{flalign*}
+ \sum\limits_{i,j,k} \biggl\{ \rho \,g\,\partial_t (z_t) \,b_t \biggr\}
+&\equiv - \sum\limits_{i,j,k} \biggl\{ \delta_k [p_w]\,\partial_t (z_t) \,e_{1t}\,e_{2t} \biggr\} &&&\\
+%
+&\equiv + \sum\limits_{i,j,k} \biggl\{ p_w\, \delta_{k+1/2} [\partial_t (z_t)] \,e_{1t}\,e_{2t} \biggr\}
+ \equiv + \sum\limits_{i,j,k} \biggl\{ p_w\, \partial_t (e_{3w}) \,e_{1t}\,e_{2t} \biggr\} &&&\\
+&\equiv + \sum\limits_{i,j,k} \biggl\{ p_w\, \partial_t (b_w) \biggr\}
+ %
+% & \equiv \sum\limits_{i,j,k} \biggl\{ \frac{1}{e_{3t}} \delta_k [p_w]\;\partial_t (z_t) \,b_w \right) \biggr\} &&&\\
+% & \equiv \sum\limits_{i,j,k} \biggl\{ p_w\;\partial_t \left( \delta_k [z_t] \right) e_{1w}\,e_{2w} \biggr\} &&&\\
+% & \equiv \sum\limits_{i,j,k} \biggl\{ p_w\;\partial_t b_w \biggr\}
+\end{flalign*}
+therefore, the balance to be satisfied is:
+\begin{flalign*}
+ \sum\limits_{i,j,k} \biggl\{ p_t\;\partial_t (b_t) \biggr\} \equiv \sum\limits_{i,j,k} \biggl\{ p_w\, \partial_t (b_w) \biggr\}
+\end{flalign*}
+which is a purely vertical balance:
+\begin{flalign*}
+ \sum\limits_{k} \biggl\{ p_t\;\partial_t (e_{3t}) \biggr\} \equiv \sum\limits_{k} \biggl\{ p_w\, \partial_t (e_{3w}) \biggr\}
+\end{flalign*}
+Defining $p_w = \overline{p_t}^{\,k+1/2}$
+
+%gm comment
+\gmcomment{
+\begin{flalign*}
+ \sum\limits_{i,j,k} \biggl\{ p_t\;\partial_t b_t \biggr\} &&&\\
+ %
+ & \equiv \sum\limits_{i,j,k} \biggl\{ \frac{1}{e_{3t}} \delta_k [p_w]\;\partial_t (z_t) \,b_w \biggr\} &&&\\
+ & \equiv \sum\limits_{i,j,k} \biggl\{ p_w\;\partial_t \left( \delta_{k+1/2} [z_t] \right) e_{1w}\,e_{2w} \biggr\} &&&\\
+ & \equiv \sum\limits_{i,j,k} \biggl\{ p_w\;\partial_t b_w \biggr\}
+\end{flalign*}
+
+
+\begin{flalign*}
+\int\limits_D \rho \, g \, \frac{\partial z }{\partial t} \;dv
+\equiv& \sum\limits_{i,j,k} \biggl\{ \frac{1}{e_{3t}} \frac{\partial e_{3t}}{\partial t} p \biggr\} \; b_t &&&\\
+\equiv& \sum\limits_{i,j,k} \biggl\{ \biggr\} \; b_t &&&\\
+\end{flalign*}
+
+%
+\begin{flalign*}
+\equiv& - \int_D \nabla \cdot \left( \rho \,\textbf {U} \right)\;g\;z\;\;dv
+ + \int\limits_D g\, \rho \; \frac{\partial z}{\partial t} \;dv &&& \\
+\end{flalign*}
+%
+}
+%end gm comment
+
+
+Note that this property strongly constrains the discrete expression of both
+the depth of $T-$points and of the term added to the pressure gradient in the
+$s$-coordinate. Nevertheless, it is almost never satisfied since a linear equation
+of state is rarely used.
+
+
+
+
+
+
+
+% ================================================================
+% Discrete Total energy Conservation : flux form
+% ================================================================
+\section{Discrete total energy conservation : flux form}
+\label{Apdx_C.1}
+
+% -------------------------------------------------------------------------------------------------------------
+% Total energy conservation
+% -------------------------------------------------------------------------------------------------------------
+\subsection{Total energy conservation}
+\label{Apdx_C_KE+PE}
+
+The discrete form of the total energy conservation, \eqref{Eq_Tot_Energy}, is given by:
+\begin{flalign*}
+\partial_t \left( \sum\limits_{i,j,k} \biggl\{ \frac{u^2}{2} \,b_u + \frac{v^2}{2}\, b_v + \rho \, g \, z_t \,b_t \biggr\} \right) &=0 \\
+\end{flalign*}
+which in flux form, it leads to:
+\begin{flalign*}
+ \sum\limits_{i,j,k} \biggl\{ \frac{u }{e_{3u}}\,\frac{\partial (e_{3u}u)}{\partial t} \,b_u
+ + \frac{v }{e_{3v}}\,\frac{\partial (e_{3v}v)}{\partial t} \,b_v \biggr\}
+& - \frac{1}{2} \sum\limits_{i,j,k} \biggl\{ \frac{u^2}{e_{3u}}\frac{\partial e_{3u} }{\partial t} \,b_u
+ + \frac{v^2}{e_{3v}}\frac{\partial e_{3v} }{\partial t} \,b_v \biggr\} \\
+&= - \sum\limits_{i,j,k} \biggl\{ \frac{1}{e_3t}\frac{\partial e_{3t} \rho}{\partial t} \, g \, z_t \,b_t \biggr\}
+ - \sum\limits_{i,j,k} \biggl\{ \rho \,g\,\frac{\partial z_t}{\partial t} \,b_t \biggr\} \\
+\end{flalign*}
+
+Substituting the discrete expression of the time derivative of the velocity either in vector invariant or in flux form,
+leads to the discrete equivalent of the
+
+
+% -------------------------------------------------------------------------------------------------------------
+% Coriolis and advection terms: flux form
+% -------------------------------------------------------------------------------------------------------------
+\subsection{Coriolis and advection terms: flux form}
+\label{Apdx_C.1.3}
+
+% -------------------------------------------------------------------------------------------------------------
+% Coriolis plus ``metric'' Term
+% -------------------------------------------------------------------------------------------------------------
+\subsubsection{Coriolis plus ``metric'' Term}
+\label{Apdx_C.1.3.1}
+
+In flux from the vorticity term reduces to a Coriolis term in which the Coriolis
+parameter has been modified to account for the ``metric'' term. This altered
+Coriolis parameter is discretised at an f-point. It is given by:
+\begin{equation*}
+f+\frac{1} {e_1 e_2 } \left( v \frac{\partial e_2 } {\partial i} - u \frac{\partial e_1 } {\partial j}\right)\;
+\equiv \;
+f+\frac{1} {e_{1f}\,e_{2f}} \left( \overline v^{\,i+1/2} \delta_{i+1/2} \left[ e_{2u} \right]
+ -\overline u^{\,j+1/2} \delta_{j+1/2} \left[ e_{1u} \right] \right)
+\end{equation*}
+
+Either the ENE or EEN scheme is then applied to obtain the vorticity term in flux form.
+It therefore conserves the total KE. The derivation is the same as for the
+vorticity term in the vector invariant form (\S\ref{Apdx_C_vor}).
+
+% -------------------------------------------------------------------------------------------------------------
+% Flux form advection
+% -------------------------------------------------------------------------------------------------------------
+\subsubsection{Flux form advection}
+\label{Apdx_C.1.3.2}
+
+The flux form operator of the momentum advection is evaluated using a
+centered second order finite difference scheme. Because of the flux form,
+the discrete operator does not contribute to the global budget of linear
+momentum. Because of the centered second order scheme, it conserves
+the horizontal kinetic energy, that is :
+
+\begin{equation} \label{Apdx_C_ADV_KE_flux}
+ - \int_D \textbf{U}_h \cdot \left( {{\begin{array} {*{20}c}
+\nabla \cdot \left( \textbf{U}\,u \right) \hfill \\
+\nabla \cdot \left( \textbf{U}\,v \right) \hfill \\ \end{array}} } \right) \;dv
+- \frac{1}{2} \int_D { {\textbf{U}_h}^2 \frac{1}{e_3} \frac{\partial e_3 }{\partial t} \;dv } =\;0
+\end{equation}
+
+Let us first consider the first term of the scalar product ($i.e.$ just the the terms
+associated with the i-component of the advection) :
+\begin{flalign*}
+& - \int_D u \cdot \nabla \cdot \left( \textbf{U}\,u \right) \; dv \\
+%
+\equiv& - \sum\limits_{i,j,k} \biggl\{ \frac{1} {b_u} \biggl(
+ \delta_{i+1/2} \left[ \overline {U}^{\,i} \;\overline u^{\,i} \right]
+ + \delta_j \left[ \overline {V}^{\,i+1/2}\;\overline u^{\,j+1/2} \right]
+ + \delta_k \left[ \overline {W}^{\,i+1/2}\;\overline u^{\,k+1/2} \right] \biggr) \; \biggr\} \, b_u \;u &&& \\
+%
+\equiv& - \sum\limits_{i,j,k}
+ \biggl\{
+ \delta_{i+1/2} \left[ \overline {U}^{\,i}\; \overline u^{\,i} \right]
+ + \delta_j \left[ \overline {V}^{\,i+1/2}\;\overline u^{\,j+1/2} \right]
+ + \delta_k \left[ \overline {W}^{\,i+12}\;\overline u^{\,k+1/2} \right]
+ \; \biggr\} \; u \\
+%
+\equiv& + \sum\limits_{i,j,k}
+ \biggl\{
+ \overline {U}^{\,i}\; \overline u^{\,i} \delta_i \left[ u \right]
+ + \overline {V}^{\,i+1/2}\; \overline u^{\,j+1/2} \delta_{j+1/2} \left[ u \right]
+ + \overline {W}^{\,i+1/2}\; \overline u^{\,k+1/2} \delta_{k+1/2} \left[ u \right] \biggr\} && \\
+%
+\equiv& + \frac{1}{2} \sum\limits_{i,j,k} \biggl\{
+ \overline{U}^{\,i} \delta_i \left[ u^2 \right]
+ + \overline{V}^{\,i+1/2} \delta_{j+/2} \left[ u^2 \right]
+ + \overline{W}^{\,i+1/2} \delta_{k+1/2} \left[ u^2 \right] \biggr\} && \\
+%
+\equiv& - \sum\limits_{i,j,k} \frac{1}{2} \bigg\{
+ U \; \delta_{i+1/2} \left[ \overline {u^2}^{\,i} \right]
+ + V \; \delta_{j+1/2} \left[ \overline {u^2}^{\,i} \right]
+ + W \; \delta_{k+1/2} \left[ \overline {u^2}^{\,i} \right] \biggr\} &&& \\
+%
+\equiv& - \sum\limits_{i,j,k} \frac{1}{2} \overline {u^2}^{\,i} \biggl\{
+ \delta_{i+1/2} \left[ U \right]
+ + \delta_{j+1/2} \left[ V \right]
+ + \delta_{k+1/2} \left[ W \right] \biggr\} &&& \\
+%
+\equiv& + \sum\limits_{i,j,k} \frac{1}{2} \overline {u^2}^{\,i}
+ \biggl\{ \left( \frac{1}{e_{3t}} \frac{\partial e_{3t}}{\partial t} \right) \; b_t \biggr\} &&& \\
+\end{flalign*}
+Applying similar manipulation applied to the second term of the scalar product
+leads to :
+\begin{equation*}
+ - \int_D \textbf{U}_h \cdot \left( {{\begin{array} {*{20}c}
+\nabla \cdot \left( \textbf{U}\,u \right) \hfill \\
+\nabla \cdot \left( \textbf{U}\,v \right) \hfill \\ \end{array}} } \right) \;dv
+\equiv + \sum\limits_{i,j,k} \frac{1}{2} \left( \overline {u^2}^{\,i} + \overline {v^2}^{\,j} \right)
+ \biggl\{ \left( \frac{1}{e_{3t}} \frac{\partial e_{3t}}{\partial t} \right) \; b_t \biggr\}
+\end{equation*}
+which is the discrete form of
+$ \frac{1}{2} \int_D u \cdot \nabla \cdot \left( \textbf{U}\,u \right) \; dv $.
+\eqref{Apdx_C_ADV_KE_flux} is thus satisfied.
+
+
+When the UBS scheme is used to evaluate the flux form momentum advection,
+the discrete operator does not contribute to the global budget of linear momentum
+(flux form). The horizontal kinetic energy is not conserved, but forced to decay
+($i.e.$ the scheme is diffusive).
+
+
+
+
+
+
+
+
+
+
+% ================================================================
+% Discrete Enstrophy Conservation
+% ================================================================
+\section{Discrete enstrophy conservation}
+\label{Apdx_C.1}
+
+
+% -------------------------------------------------------------------------------------------------------------
+% Vorticity Term with ENS scheme
+% -------------------------------------------------------------------------------------------------------------
+\subsubsection{Vorticity Term with ENS scheme (\np{ln\_dynvor\_ens}=.true.)}
+\label{Apdx_C_vorENS}
+
+In the ENS scheme, the vorticity term is descretized as follows:
+\begin{equation} \label{Eq_dynvor_ens}
+\left\{ \begin{aligned}
++\frac{1}{e_{1u}} & \overline{q}^{\,i} & {\overline{ \overline{\left( e_{1v}\,e_{3v}\; v \right) } } }^{\,i, j+1/2} \\
+- \frac{1}{e_{2v}} & \overline{q}^{\,j} & {\overline{ \overline{\left( e_{2u}\,e_{3u}\; u \right) } } }^{\,i+1/2, j}
+\end{aligned} \right.
+\end{equation}
+
+The scheme does not allow but the conservation of the total kinetic energy but the conservation
+of $q^2$, the potential enstrophy for a horizontally non-divergent flow ($i.e.$ when $\chi$=$0$).
+Indeed, using the symmetry or skew symmetry properties of the operators (Eqs \eqref{DOM_mi_adj}
+and \eqref{DOM_di_adj}), it can be shown that:
+\begin{equation} \label{Apdx_C_1.1}
+\int_D {q\,\;{\textbf{k}}\cdot \frac{1} {e_3} \nabla \times \left( {e_3 \, q \;{\textbf{k}}\times {\textbf{U}}_h } \right)\;dv} \equiv 0
+\end{equation}
+where $dv=e_1\,e_2\,e_3 \; di\,dj\,dk$ is the volume element. Indeed, using
+\eqref{Eq_dynvor_ens}, the discrete form of the right hand side of \eqref{Apdx_C_1.1}
+can be transformed as follow:
+\begin{flalign*}
+&\int_D q \,\; \textbf{k} \cdot \frac{1} {e_3 } \nabla \times
+ \left( e_3 \, q \; \textbf{k} \times \textbf{U}_h \right)\; dv \\
+%
+& \qquad {\begin{array}{*{20}l}
+&\equiv \sum\limits_{i,j,k}
+q \ \left\{ \delta_{i+1/2} \left[ - \,\overline {q}^{\,i}\; \overline{\overline U }^{\,i,j+1/ 2} \right]
+ - \delta_{j+1/2} \left[ \overline {q}^{\,j}\; \overline{\overline V }^{\,i+1/2, j} \right] \right\} \\
+%
+&\equiv \sum\limits_{i,j,k}
+ \left\{ \delta_i [q] \; \overline{q}^{\,i} \; \overline{ \overline U }^{\,i,j+1/2}
+ + \delta_j [q] \; \overline{q}^{\,j} \; \overline{\overline V }^{\,i+1/2,j} \right\} && \\
+%
+&\equiv \,\frac{1} {2} \sum\limits_{i,j,k}
+ \left\{ \delta_i \left[ q^2 \right] \; \overline{\overline U }^{\,i,j+1/2}
+ + \delta_j \left[ q^2 \right] \; \overline{\overline V }^{\,i+1/2,j} \right\} && \\
+%
+&\equiv - \frac{1} {2} \sum\limits_{i,j,k} q^2 \;
+ \left\{ \delta_{i+1/2} \left[ \overline{\overline{ U }}^{\,i,j+1/2} \right]
+ + \delta_{j+1/2} \left[ \overline{\overline{ V }}^{\,i+1/2,j} \right] \right\} && \\
+\end{array} }
+%
+\allowdisplaybreaks
+\intertext{ Since $\overline {\;\cdot \;} $ and $\delta $ operators commute: $\delta_{i+1/2}
+\left[ {\overline a^{\,i}} \right] = \overline {\delta_i \left[ a \right]}^{\,i+1/2}$,
+and introducing the horizontal divergence $\chi $, it becomes: }
+\allowdisplaybreaks
+%
+& \qquad {\begin{array}{*{20}l}
+&\equiv \sum\limits_{i,j,k} - \frac{1} {2} q^2 \; \overline{\overline{ e_{1t}\,e_{2t}\,e_{3t}^{}\, \chi}}^{\,i+1/2,j+1/2}
+\quad \equiv 0 &&
+\end{array} }
+\end{flalign*}
+The later equality is obtain only when the flow is horizontally non-divergent, $i.e.$ $\chi$=$0$.
+
+
+% -------------------------------------------------------------------------------------------------------------
+% Vorticity Term with EEN scheme
+% -------------------------------------------------------------------------------------------------------------
+\subsubsection{Vorticity Term with EEN scheme (\np{ln\_dynvor\_een}=.true.)}
+\label{Apdx_C_vorEEN}
+
+With the EEN scheme, the vorticity terms are represented as:
+\begin{equation} \label{Eq_dynvor_een}
+\left\{ { \begin{aligned}
+ +q\,e_3 \, v &\equiv +\frac{1}{e_{1u} } \sum_{\substack{i_p,\,k_p}}
+ {^{i+1/2-i_p}_j} \mathbb{Q}^{i_p}_{j_p} \left( e_{1v} e_{3v} \ v \right)^{i+i_p-1/2}_{j+j_p} \\
+ - q\,e_3 \, u &\equiv -\frac{1}{e_{2v} } \sum_{\substack{i_p,\,k_p}}
+ {^i_{j+1/2-j_p}} \mathbb{Q}^{i_p}_{j_p} \left( e_{2u} e_{3u} \ u \right)^{i+i_p}_{j+j_p-1/2} \\
+\end{aligned} } \right.
+\end{equation}
+where the indices $i_p$ and $k_p$ take the following value:
+$i_p = -1/2$ or $1/2$ and $j_p = -1/2$ or $1/2$,
+and the vorticity triads, ${^i_j}\mathbb{Q}^{i_p}_{j_p}$, defined at $T$-point, are given by:
+\begin{equation} \label{Q_triads}
+_i^j \mathbb{Q}^{i_p}_{j_p}
+= \frac{1}{12} \ \left( q^{i-i_p}_{j+j_p} + q^{i+j_p}_{j+i_p} + q^{i+i_p}_{j-j_p} \right)
+\end{equation}
+
+
+This formulation does conserve the potential enstrophy for a horizontally non-divergent flow ($i.e.$ $\chi=0$).
+
+Let consider one of the vorticity triad, for example ${^{i}_j}\mathbb{Q}^{+1/2}_{+1/2} $,
+similar manipulation can be done for the 3 others. The discrete form of the right hand
+side of \eqref{Apdx_C_1.1} applied to this triad only can be transformed as follow:
+
+\begin{flalign*}
+&\int_D {q\,\;{\textbf{k}}\cdot \frac{1} {e_3} \nabla \times \left( {e_3 \, q \;{\textbf{k}}\times {\textbf{U}}_h } \right)\;dv} \\
+%
+\equiv& \sum\limits_{i,j,k}
+ {q} \ \biggl\{ \;\;
+ \delta_{i+1/2} \left[ -\, {{^i_j}\mathbb{Q}^{+1/2}_{+1/2} \; U^{i+1/2}_{j}} \right]
+ - \delta_{j+1/2} \left[ {{^i_j}\mathbb{Q}^{+1/2}_{+1/2} \; V^{i}_{j+1/2}} \right]
+ \;\;\biggr\} && \\
+%
+\equiv& \sum\limits_{i,j,k}
+ \biggl\{ \delta_i [q] \ {{^i_j}\mathbb{Q}^{+1/2}_{+1/2} \; U^{i+1/2}_{j}}
+ + \delta_j [q] \ {{^i_j}\mathbb{Q}^{+1/2}_{+1/2} \; V^{i}_{j+1/2}} \biggr\}
+ && \\
+%
+... & &&\\
+&Demonstation \ to \ be \ done... &&\\
+... & &&\\
+%
+\equiv& \frac{1} {2} \sum\limits_{i,j,k}
+ \biggl\{ \delta_i \Bigl[ \left( {{^i_j}\mathbb{Q}^{+1/2}_{+1/2}} \right)^2 \Bigr]\;
+ \overline{\overline {U}}^{\,i,j+1/2}
+ + \delta_j \Bigl[ \left( {{^i_j}\mathbb{Q}^{+1/2}_{+1/2}} \right)^2 \Bigr]\;
+ \overline{\overline {V}}^{\,i+1/2,j}
+ \biggr\}
+ && \\
+%
+\equiv& - \frac{1} {2} \sum\limits_{i,j,k} \left( {{^i_j}\mathbb{Q}^{+1/2}_{+1/2}} \right)^2\;
+ \biggl\{ \delta_{i+1/2}
+ \left[ \overline{\overline {U}}^{\,i,j+1/2} \right]
+ + \delta_{j+1/2}
+ \left[ \overline{\overline {V}}^{\,i+1/2,j} \right]
+ \biggr\} && \\
+%
+\equiv& \sum\limits_{i,j,k} - \frac{1} {2} \left( {{^i_j}\mathbb{Q}^{+1/2}_{+1/2}} \right)^2
+ \; \overline{\overline{ b_t^{}\, \chi}}^{\,i+1/2,\,j+1/2} &&\\
+%
+\ \ \equiv& \ 0 &&\\
+\end{flalign*}
+
+
+
+
+
+% ================================================================
+% Conservation Properties on Tracers
+% ================================================================
+\section{Conservation Properties on Tracers}
+\label{Apdx_C.2}
+
+
+All the numerical schemes used in NEMO are written such that the tracer content
+is conserved by the internal dynamics and physics (equations in flux form).
+For advection, only the CEN2 scheme ($i.e.$ $2^{nd}$ order finite different scheme)
+conserves the global variance of tracer. Nevertheless the other schemes ensure
+that the global variance decreases ($i.e.$ they are at least slightly diffusive).
+For diffusion, all the schemes ensure the decrease of the total tracer variance,
+except the iso-neutral operator. There is generally no strict conservation of mass,
+as the equation of state is non linear with respect to $T$ and $S$. In practice,
+the mass is conserved to a very high accuracy.
+% -------------------------------------------------------------------------------------------------------------
+% Advection Term
+% -------------------------------------------------------------------------------------------------------------
+\subsection{Advection Term}
+\label{Apdx_C.2.1}
+
+conservation of a tracer, $T$:
+\begin{equation*}
+\frac{\partial }{\partial t} \left( \int_D {T\;dv} \right)
+= \int_D { \frac{1}{e_3}\frac{\partial \left( e_3 \, T \right)}{\partial t} \;dv }=0
+\end{equation*}
+
+conservation of its variance:
+\begin{flalign*}
+\frac{\partial }{\partial t} \left( \int_D {\frac{1}{2} T^2\;dv} \right)
+=& \int_D { \frac{1}{e_3} Q \frac{\partial \left( e_3 \, T \right) }{\partial t} \;dv }
+- \frac{1}{2} \int_D { T^2 \frac{1}{e_3} \frac{\partial e_3 }{\partial t} \;dv }
+\end{flalign*}
+
+
+Whatever the advection scheme considered it conserves of the tracer content as all
+the scheme are written in flux form. Indeed, let $T$ be the tracer and $\tau_u$, $\tau_v$,
+and $\tau_w$ its interpolated values at velocity point (whatever the interpolation is),
+the conservation of the tracer content due to the advection tendency is obtained as follows:
+\begin{flalign*}
+&\int_D { \frac{1}{e_3}\frac{\partial \left( e_3 \, T \right)}{\partial t} \;dv } = - \int_D \nabla \cdot \left( T \textbf{U} \right)\;dv &&&\\
+&\equiv - \sum\limits_{i,j,k} \biggl\{
+ \frac{1} {b_t} \left( \delta_i \left[ U \;\tau_u \right]
+ + \delta_j \left[ V \;\tau_v \right] \right)
+ + \frac{1} {e_{3t}} \delta_k \left[ w\;\tau_w \right] \biggl\} b_t &&&\\
+%
+&\equiv - \sum\limits_{i,j,k} \left\{
+ \delta_i \left[ U \;\tau_u \right]
+ + \delta_j \left[ V \;\tau_v \right]
+ + \delta_k \left[ W \;\tau_w \right] \right\} && \\
+&\equiv 0 &&&
+\end{flalign*}
+
+The conservation of the variance of tracer due to the advection tendency
+can be achieved only with the CEN2 scheme, $i.e.$ when
+$\tau_u= \overline T^{\,i+1/2}$, $\tau_v= \overline T^{\,j+1/2}$, and $\tau_w= \overline T^{\,k+1/2}$.
+It can be demonstarted as follows:
+\begin{flalign*}
+&\int_D { \frac{1}{e_3} Q \frac{\partial \left( e_3 \, T \right) }{\partial t} \;dv }
+= - \int\limits_D \tau\;\nabla \cdot \left( T\; \textbf{U} \right)\;dv &&&\\
+\equiv& - \sum\limits_{i,j,k} T\;
+ \left\{
+ \delta_i \left[ U \overline T^{\,i+1/2} \right]
+ + \delta_j \left[ V \overline T^{\,j+1/2} \right]
+ + \delta_k \left[ W \overline T^{\,k+1/2} \right] \right\} && \\
+\equiv& + \sum\limits_{i,j,k}
+ \left\{ U \overline T^{\,i+1/2} \,\delta_{i+1/2} \left[ T \right]
+ + V \overline T^{\,j+1/2} \;\delta_{j+1/2} \left[ T \right]
+ + W \overline T^{\,k+1/2}\;\delta_{k+1/2} \left[ T \right] \right\} &&\\
+\equiv& + \frac{1} {2} \sum\limits_{i,j,k}
+ \Bigl\{ U \;\delta_{i+1/2} \left[ T^2 \right]
+ + V \;\delta_{j+1/2} \left[ T^2 \right]
+ + W \;\delta_{k+1/2} \left[ T^2 \right] \Bigr\} && \\
+\equiv& - \frac{1} {2} \sum\limits_{i,j,k} T^2
+ \Bigl\{ \delta_i \left[ U \right]
+ + \delta_j \left[ V \right]
+ + \delta_k \left[ W \right] \Bigr\} &&& \\
+\equiv& + \frac{1} {2} \sum\limits_{i,j,k} T^2
+ \Bigl\{ \frac{1}{e_{3t}} \frac{\partial e_{3t}\,T }{\partial t} \Bigr\} &&& \\
+\end{flalign*}
+which is the discrete form of $ \frac{1}{2} \int_D { T^2 \frac{1}{e_3} \frac{\partial e_3 }{\partial t} \;dv }$.
+
+% ================================================================
+% Conservation Properties on Lateral Momentum Physics
+% ================================================================
+\section{Conservation Properties on Lateral Momentum Physics}
+\label{Apdx_dynldf_properties}
+
+
+The discrete formulation of the horizontal diffusion of momentum ensures the
+conservation of potential vorticity and the horizontal divergence, and the
+dissipation of the square of these quantities (i.e. enstrophy and the
+variance of the horizontal divergence) as well as the dissipation of the
+horizontal kinetic energy. In particular, when the eddy coefficients are
+horizontally uniform, it ensures a complete separation of vorticity and
+horizontal divergence fields, so that diffusion (dissipation) of vorticity
+(enstrophy) does not generate horizontal divergence (variance of the
+horizontal divergence) and \textit{vice versa}.
+
+These properties of the horizontal diffusion operator are a direct consequence
+of properties \eqref{Eq_DOM_curl_grad} and \eqref{Eq_DOM_div_curl}.
+When the vertical curl of the horizontal diffusion of momentum (discrete sense)
+is taken, the term associated with the horizontal gradient of the divergence is
+locally zero.
+
+% -------------------------------------------------------------------------------------------------------------
+% Conservation of Potential Vorticity
+% -------------------------------------------------------------------------------------------------------------
+\subsection{Conservation of Potential Vorticity}
+\label{Apdx_C.3.1}
+
+The lateral momentum diffusion term conserves the potential vorticity :
+\begin{flalign*}
+&\int \limits_D \frac{1} {e_3 } \textbf{k} \cdot \nabla \times
+ \Bigl[ \nabla_h \left( A^{\,lm}\;\chi \right)
+ - \nabla_h \times \left( A^{\,lm}\;\zeta \; \textbf{k} \right) \Bigr]\;dv = 0
+\end{flalign*}
+%%%%%%%%%% recheck here.... (gm)
+\begin{flalign*}
+= \int \limits_D -\frac{1} {e_3 } \textbf{k} \cdot \nabla \times
+ \Bigl[ \nabla_h \times \left( A^{\,lm}\;\zeta \; \textbf{k} \right) \Bigr]\;dv &&& \\
+\end{flalign*}
+\begin{flalign*}
+\equiv& \sum\limits_{i,j}
+ \left\{
+ \delta_{i+1/2}
+ \left[
+ \frac {e_{2v}} {e_{1v}\,e_{3v}} \delta_i
+ \left[ A_f^{\,lm} e_{3f} \zeta \right]
+ \right]
+ + \delta_{j+1/2}
+ \left[
+ \frac {e_{1u}} {e_{2u}\,e_{3u}} \delta_j
+ \left[ A_f^{\,lm} e_{3f} \zeta \right]
+ \right]
+ \right\}
+ && \\
+%
+\intertext{Using \eqref{DOM_di_adj}, it follows:}
+%
+\equiv& \sum\limits_{i,j,k}
+ -\,\left\{
+ \frac{e_{2v}} {e_{1v}\,e_{3v}} \delta_i
+ \left[ A_f^{\,lm} e_{3f} \zeta \right]\;\delta_i \left[ 1\right]
+ + \frac{e_{1u}} {e_{2u}\,e_{3u}} \delta_j
+ \left[ A_f^{\,lm} e_{3f} \zeta \right]\;\delta_j \left[ 1\right]
+ \right\} \quad \equiv 0
+ && \\
+\end{flalign*}
+
+% -------------------------------------------------------------------------------------------------------------
+% Dissipation of Horizontal Kinetic Energy
+% -------------------------------------------------------------------------------------------------------------
+\subsection{Dissipation of Horizontal Kinetic Energy}
+\label{Apdx_C.3.2}
+
+
+The lateral momentum diffusion term dissipates the horizontal kinetic energy:
+%\begin{flalign*}
+\begin{equation*}
+\begin{split}
+\int_D \textbf{U}_h \cdot
+ \left[ \nabla_h \right. & \left. \left( A^{\,lm}\;\chi \right)
+ - \nabla_h \times \left( A^{\,lm}\;\zeta \;\textbf{k} \right) \right] \; dv \\
+\\ %%%
+\equiv& \sum\limits_{i,j,k}
+ \left\{
+ \frac{1} {e_{1u}} \delta_{i+1/2} \left[ A_T^{\,lm} \chi \right]
+ - \frac{1} {e_{2u}\,e_{3u}} \delta_j \left[ A_f^{\,lm} e_{3f} \zeta \right]
+ \right\} \; e_{1u}\,e_{2u}\,e_{3u} \;u \\
+&\;\; + \left\{
+ \frac{1} {e_{2u}} \delta_{j+1/2} \left[ A_T^{\,lm} \chi \right]
+ + \frac{1} {e_{1v}\,e_{3v}} \delta_i \left[ A_f^{\,lm} e_{3f} \zeta \right]
+ \right\} \; e_{1v}\,e_{2u}\,e_{3v} \;v \qquad \\
+\\ %%%
+\equiv& \sum\limits_{i,j,k}
+ \Bigl\{
+ e_{2u}\,e_{3u} \;u\; \delta_{i+1/2} \left[ A_T^{\,lm} \chi \right]
+ - e_{1u} \;u\; \delta_j \left[ A_f^{\,lm} e_{3f} \zeta \right]
+ \Bigl\}
+ \\
+&\;\; + \Bigl\{
+ e_{1v}\,e_{3v} \;v\; \delta_{j+1/2} \left[ A_T^{\,lm} \chi \right]
+ + e_{2v} \;v\; \delta_i \left[ A_f^{\,lm} e_{3f} \zeta \right]
+ \Bigl\} \\
+\\ %%%
+\equiv& \sum\limits_{i,j,k}
+ - \Bigl(
+ \delta_i \left[ e_{2u}\,e_{3u} \;u \right]
+ + \delta_j \left[ e_{1v}\,e_{3v} \;v \right]
+ \Bigr) \; A_T^{\,lm} \chi \\
+&\;\; - \Bigl(
+ \delta_{i+1/2} \left[ e_{2v} \;v \right]
+ - \delta_{j+1/2} \left[ e_{1u} \;u \right]
+ \Bigr)\; A_f^{\,lm} e_{3f} \zeta \\
+\\ %%%
+\equiv& \sum\limits_{i,j,k}
+ - A_T^{\,lm} \,\chi^2 \;e_{1t}\,e_{2t}\,e_{3t}
+ - A_f ^{\,lm} \,\zeta^2 \;e_{1f }\,e_{2f }\,e_{3f}
+\quad \leq 0 \\
+\end{split}
+\end{equation*}
+
+% -------------------------------------------------------------------------------------------------------------
+% Dissipation of Enstrophy
+% -------------------------------------------------------------------------------------------------------------
+\subsection{Dissipation of Enstrophy}
+\label{Apdx_C.3.3}
+
+
+The lateral momentum diffusion term dissipates the enstrophy when the eddy
+coefficients are horizontally uniform:
+\begin{flalign*}
+&\int\limits_D \zeta \; \textbf{k} \cdot \nabla \times
+ \left[ \nabla_h \left( A^{\,lm}\;\chi \right)
+ - \nabla_h \times \left( A^{\,lm}\;\zeta \; \textbf{k} \right) \right]\;dv &&&\\
+&= A^{\,lm} \int \limits_D \zeta \textbf{k} \cdot \nabla \times
+ \left[ \nabla_h \times \left( \zeta \; \textbf{k} \right) \right]\;dv &&&\\
+&\equiv A^{\,lm} \sum\limits_{i,j,k} \zeta \;e_{3f}
+ \left\{ \delta_{i+1/2} \left[ \frac{e_{2v}} {e_{1v}\,e_{3v}} \delta_i \left[ e_{3f} \zeta \right] \right]
+ + \delta_{j+1/2} \left[ \frac{e_{1u}} {e_{2u}\,e_{3u}} \delta_j \left[ e_{3f} \zeta \right] \right] \right\} &&&\\
+%
+\intertext{Using \eqref{DOM_di_adj}, it follows:}
+%
+&\equiv - A^{\,lm} \sum\limits_{i,j,k}
+ \left\{ \left( \frac{1} {e_{1v}\,e_{3v}} \delta_i \left[ e_{3f} \zeta \right] \right)^2 b_v
+ + \left( \frac{1} {e_{2u}\,e_{3u}} \delta_j \left[ e_{3f} \zeta \right] \right)^2 b_u \right\} &&&\\
+& \leq \;0 &&&\\
+\end{flalign*}
+
+% -------------------------------------------------------------------------------------------------------------
+% Conservation of Horizontal Divergence
+% -------------------------------------------------------------------------------------------------------------
+\subsection{Conservation of Horizontal Divergence}
+\label{Apdx_C.3.4}
+
+When the horizontal divergence of the horizontal diffusion of momentum
+(discrete sense) is taken, the term associated with the vertical curl of the
+vorticity is zero locally, due to (!!! II.1.8 !!!!!). The resulting term conserves the
+$\chi$ and dissipates $\chi^2$ when the eddy coefficients are
+horizontally uniform.
+\begin{flalign*}
+& \int\limits_D \nabla_h \cdot
+ \Bigl[ \nabla_h \left( A^{\,lm}\;\chi \right)
+ - \nabla_h \times \left( A^{\,lm}\;\zeta \;\textbf{k} \right) \Bigr] dv
+= \int\limits_D \nabla_h \cdot \nabla_h \left( A^{\,lm}\;\chi \right) dv &&&\\
+%
+&\equiv \sum\limits_{i,j,k}
+ \left\{ \delta_i \left[ A_u^{\,lm} \frac{e_{2u}\,e_{3u}} {e_{1u}} \delta_{i+1/2} \left[ \chi \right] \right]
+ + \delta_j \left[ A_v^{\,lm} \frac{e_{1v}\,e_{3v}} {e_{2v}} \delta_{j+1/2} \left[ \chi \right] \right] \right\} &&&\\
+%
+\intertext{Using \eqref{DOM_di_adj}, it follows:}
+%
+&\equiv \sum\limits_{i,j,k}
+ - \left\{ \frac{e_{2u}\,e_{3u}} {e_{1u}} A_u^{\,lm} \delta_{i+1/2} \left[ \chi \right] \delta_{i+1/2} \left[ 1 \right]
+ + \frac{e_{1v}\,e_{3v}} {e_{2v}} A_v^{\,lm} \delta_{j+1/2} \left[ \chi \right] \delta_{j+1/2} \left[ 1 \right] \right\}
+ \qquad \equiv 0 &&& \\
+\end{flalign*}
+
+% -------------------------------------------------------------------------------------------------------------
+% Dissipation of Horizontal Divergence Variance
+% -------------------------------------------------------------------------------------------------------------
+\subsection{Dissipation of Horizontal Divergence Variance}
+\label{Apdx_C.3.5}
+
+\begin{flalign*}
+&\int\limits_D \chi \;\nabla_h \cdot
+ \left[ \nabla_h \left( A^{\,lm}\;\chi \right)
+ - \nabla_h \times \left( A^{\,lm}\;\zeta \;\textbf{k} \right) \right]\; dv
+ = A^{\,lm} \int\limits_D \chi \;\nabla_h \cdot \nabla_h \left( \chi \right)\; dv &&&\\
+%
+&\equiv A^{\,lm} \sum\limits_{i,j,k} \frac{1} {e_{1t}\,e_{2t}\,e_{3t}} \chi
+ \left\{
+ \delta_i \left[ \frac{e_{2u}\,e_{3u}} {e_{1u}} \delta_{i+1/2} \left[ \chi \right] \right]
+ + \delta_j \left[ \frac{e_{1v}\,e_{3v}} {e_{2v}} \delta_{j+1/2} \left[ \chi \right] \right]
+ \right\} \; e_{1t}\,e_{2t}\,e_{3t} &&&\\
+%
+\intertext{Using \eqref{DOM_di_adj}, it turns out to be:}
+%
+&\equiv - A^{\,lm} \sum\limits_{i,j,k}
+ \left\{ \left( \frac{1} {e_{1u}} \delta_{i+1/2} \left[ \chi \right] \right)^2 b_u
+ + \left( \frac{1} {e_{2v}} \delta_{j+1/2} \left[ \chi \right] \right)^2 b_v \right\} \; &&&\\
+%
+&\leq 0 &&&\\
+\end{flalign*}
+
+% ================================================================
+% Conservation Properties on Vertical Momentum Physics
+% ================================================================
+\section{Conservation Properties on Vertical Momentum Physics}
+\label{Apdx_C_4}
+
+
+As for the lateral momentum physics, the continuous form of the vertical diffusion
+of momentum satisfies several integral constraints. The first two are associated
+with the conservation of momentum and the dissipation of horizontal kinetic energy:
+\begin{align*}
+ \int\limits_D \frac{1} {e_3 }\; \frac{\partial } {\partial k}
+ \left( \frac{A^{\,vm}} {e_3 }\; \frac{\partial \textbf{U}_h } {\partial k} \right)\; dv
+ \qquad \quad &= \vec{\textbf{0}} \\
+%
+\intertext{and}
+%
+\int\limits_D
+ \textbf{U}_h \cdot \frac{1} {e_3 }\; \frac{\partial } {\partial k}
+ \left( \frac{A^{\,vm}} {e_3 }\; \frac{\partial \textbf{U}_h } {\partial k} \right)\; dv \quad &\leq 0 \\
+\end{align*}
+The first property is obvious. The second results from:
+
+\begin{flalign*}
+\int\limits_D
+ \textbf{U}_h \cdot \frac{1} {e_3 }\; \frac{\partial } {\partial k}
+ \left( \frac{A^{\,vm}} {e_3 }\; \frac{\partial \textbf{U}_h } {\partial k} \right)\;dv &&&\\
+\end{flalign*}
+\begin{flalign*}
+&\equiv \sum\limits_{i,j,k}
+ \left(
+ u\; \delta_k \left[ \frac{A_u^{\,vm}} {e_{3uw}} \delta_{k+1/2} \left[ u \right] \right]\; e_{1u}\,e_{2u}
+ + v\; \delta_k \left[ \frac{A_v^{\,vm}} {e_{3vw}} \delta_{k+1/2} \left[ v \right] \right]\; e_{1v}\,e_{2v} \right) &&&\\
+%
+\intertext{since the horizontal scale factor does not depend on $k$, it follows:}
+%
+&\equiv - \sum\limits_{i,j,k}
+ \left( \frac{A_u^{\,vm}} {e_{3uw}} \left( \delta_{k+1/2} \left[ u \right] \right)^2\; e_{1u}\,e_{2u}
+ + \frac{A_v^{\,vm}} {e_{3vw}} \left( \delta_{k+1/2} \left[ v \right] \right)^2\; e_{1v}\,e_{2v} \right)
+\quad \leq 0 &&&\\
+\end{flalign*}
+
+The vorticity is also conserved. Indeed:
+\begin{flalign*}
+\int \limits_D
+ \frac{1} {e_3 } \textbf{k} \cdot \nabla \times
+ \left( \frac{1} {e_3 }\; \frac{\partial } {\partial k} \left(
+ \frac{A^{\,vm}} {e_3}\; \frac{\partial \textbf{U}_h } {\partial k}
+ \right) \right)\; dv &&&\\
+\end{flalign*}
+\begin{flalign*}
+\equiv \sum\limits_{i,j,k} \frac{1} {e_{3f}}\; \frac{1} {e_{1f}\,e_{2f}}
+ \bigg\{ \biggr. \quad
+ \delta_{i+1/2}
+ &\left( \frac{e_{2v}} {e_{3v}} \delta_k \left[ \frac{1} {e_{3vw}} \delta_{k+1/2} \left[ v \right] \right] \right) &&\\
+ \biggl.
+ - \delta_{j+1/2}
+ &\left( \frac{e_{1u}} {e_{3u}} \delta_k \left[ \frac{1} {e_{3uw}}\delta_{k+1/2} \left[ u \right] \right] \right)
+ \biggr\} \;
+ e_{1f}\,e_{2f}\,e_{3f} \; \equiv 0 && \\
+\end{flalign*}
+If the vertical diffusion coefficient is uniform over the whole domain, the
+enstrophy is dissipated, $i.e.$
+\begin{flalign*}
+\int\limits_D \zeta \, \textbf{k} \cdot \nabla \times
+ \left( \frac{1} {e_3}\; \frac{\partial } {\partial k}
+ \left( \frac{A^{\,vm}} {e_3 }\; \frac{\partial \textbf{U}_h } {\partial k} \right) \right)\; dv = 0 &&&\\
+\end{flalign*}
+This property is only satisfied in $z$-coordinates:
+
+\begin{flalign*}
+\int\limits_D \zeta \, \textbf{k} \cdot \nabla \times
+ \left( \frac{1} {e_3}\; \frac{\partial } {\partial k}
+ \left( \frac{A^{\,vm}} {e_3 }\; \frac{\partial \textbf{U}_h } {\partial k} \right) \right)\; dv &&& \\
+\end{flalign*}
+\begin{flalign*}
+\equiv \sum\limits_{i,j,k} \zeta \;e_{3f} \;
+ \biggl\{ \biggr. \quad
+ \delta_{i+1/2}
+ &\left( \frac{e_{2v}} {e_{3v}} \delta_k \left[ \frac{A_v^{\,vm}} {e_{3vw}} \delta_{k+1/2}[v] \right] \right) &&\\
+ - \delta_{j+1/2}
+ &\biggl.
+ \left( \frac{e_{1u}} {e_{3u}} \delta_k \left[ \frac{A_u^{\,vm}} {e_{3uw}} \delta_{k+1/2} [u] \right] \right) \biggr\} &&\\
+\end{flalign*}
+\begin{flalign*}
+\equiv \sum\limits_{i,j,k} \zeta \;e_{3f}
+ \biggl\{ \biggr. \quad
+ \frac{1} {e_{3v}} \delta_k
+ &\left[ \frac{A_v^{\,vm}} {e_{3vw}} \delta_{k+1/2} \left[ \delta_{i+1/2} \left[ e_{2v}\,v \right] \right] \right] &&\\
+ \biggl.
+ - \frac{1} {e_{3u}} \delta_k
+ &\left[ \frac{A_u^{\,vm}} {e_{3uw}} \delta_{k+1/2} \left[ \delta_{j+1/2} \left[ e_{1u}\,u \right] \right] \right] \biggr\} &&\\
+\end{flalign*}
+Using the fact that the vertical diffusion coefficients are uniform, and that in
+$z$-coordinate, the vertical scale factors do not depend on $i$ and $j$ so
+that: $e_{3f} =e_{3u} =e_{3v} =e_{3t} $ and $e_{3w} =e_{3uw} =e_{3vw} $,
+it follows:
+\begin{flalign*}
+\equiv A^{\,vm} \sum\limits_{i,j,k} \zeta \;\delta_k
+ \left[ \frac{1} {e_{3w}} \delta_{k+1/2} \Bigl[ \delta_{i+1/2} \left[ e_{2v}\,v \right]
+ - \delta_{j+1/ 2} \left[ e_{1u}\,u \right] \Bigr] \right] &&&\\
+\end{flalign*}
+\begin{flalign*}
+\equiv - A^{\,vm} \sum\limits_{i,j,k} \frac{1} {e_{3w}}
+ \left( \delta_{k+1/2} \left[ \zeta \right] \right)^2 \; e_{1f}\,e_{2f} \; \leq 0 &&&\\
+\end{flalign*}
+Similarly, the horizontal divergence is obviously conserved:
+
+\begin{flalign*}
+\int\limits_D \nabla \cdot
+\left( \frac{1} {e_3 }\; \frac{\partial } {\partial k}
+ \left( \frac{A^{\,vm}} {e_3 }\; \frac{\partial \textbf{U}_h } {\partial k} \right) \right)\; dv = 0 &&&\\
+\end{flalign*}
+and the square of the horizontal divergence decreases ($i.e.$ the horizontal
+divergence is dissipated) if the vertical diffusion coefficient is uniform over the
+whole domain:
+
+\begin{flalign*}
+\int\limits_D \chi \;\nabla \cdot
+\left( \frac{1} {e_3 }\; \frac{\partial } {\partial k}
+ \left( \frac{A^{\,vm}} {e_3 }\; \frac{\partial \textbf{U}_h } {\partial k} \right) \right)\; dv = 0 &&&\\
+\end{flalign*}
+This property is only satisfied in the $z$-coordinate:
+\begin{flalign*}
+\int\limits_D \chi \;\nabla \cdot
+\left( \frac{1} {e_3 }\; \frac{\partial } {\partial k}
+ \left( \frac{A^{\,vm}} {e_3 }\; \frac{\partial \textbf{U}_h } {\partial k} \right) \right)\; dv &&&\\
+\end{flalign*}
+\begin{flalign*}
+\equiv \sum\limits_{i,j,k} \frac{\chi } {e_{1t}\,e_{2t}}
+ \biggl\{ \Biggr. \quad
+ \delta_{i+1/2}
+ &\left( \frac{e_{2u}} {e_{3u}} \delta_k
+ \left[ \frac{A_u^{\,vm}} {e_{3uw}} \delta_{k+1/2} [u] \right] \right) &&\\
+ \Biggl.
+ + \delta_{j+1/2}
+ &\left( \frac{e_{1v}} {e_{3v}} \delta_k
+ \left[ \frac{A_v^{\,vm}} {e_{3vw}} \delta_{k+1/2} [v] \right] \right)
+ \Biggr\} \; e_{1t}\,e_{2t}\,e_{3t} &&\\
+\end{flalign*}
+
+\begin{flalign*}
+\equiv A^{\,vm} \sum\limits_{i,j,k} \chi \,
+ \biggl\{ \biggr. \quad
+ \delta_{i+1/2}
+ &\left(
+ \delta_k \left[
+ \frac{1} {e_{3uw}} \delta_{k+1/2} \left[ e_{2u}\,u \right] \right] \right) && \\
+ \biggl.
+ + \delta_{j+1/2}
+ &\left( \delta_k \left[
+ \frac{1} {e_{3vw}} \delta_{k+1/2} \left[ e_{1v}\,v \right] \right] \right) \biggr\} && \\
+\end{flalign*}
+
+\begin{flalign*}
+\equiv -A^{\,vm} \sum\limits_{i,j,k}
+\frac{\delta_{k+1/2} \left[ \chi \right]} {e_{3w}}\; \biggl\{
+ \delta_{k+1/2} \Bigl[
+ \delta_{i+1/2} \left[ e_{2u}\,u \right]
+ + \delta_{j+1/2} \left[ e_{1v}\,v \right] \Bigr] \biggr\} &&&\\
+\end{flalign*}
+
+\begin{flalign*}
+\equiv -A^{\,vm} \sum\limits_{i,j,k}
+ \frac{1} {e_{3w}} \delta_{k+1/2} \left[ \chi \right]\; \delta_{k+1/2} \left[ e_{1t}\,e_{2t} \;\chi \right] &&&\\
+\end{flalign*}
+
+\begin{flalign*}
+\equiv -A^{\,vm} \sum\limits_{i,j,k}
+\frac{e_{1t}\,e_{2t}} {e_{3w}}\; \left( \delta_{k+1/2} \left[ \chi \right] \right)^2 \quad \equiv 0 &&&\\
+\end{flalign*}
+
+% ================================================================
+% Conservation Properties on Tracer Physics
+% ================================================================
+\section{Conservation Properties on Tracer Physics}
+\label{Apdx_C.5}
+
+The numerical schemes used for tracer subgridscale physics are written such
+that the heat and salt contents are conserved (equations in flux form, second
+order centered finite differences). Since a flux form is used to compute the
+temperature and salinity, the quadratic form of these quantities (i.e. their variance)
+globally tends to diminish. As for the advection term, there is generally no strict
+conservation of mass, even if in practice the mass is conserved to a very high
+accuracy.
+
+% -------------------------------------------------------------------------------------------------------------
+% Conservation of Tracers
+% -------------------------------------------------------------------------------------------------------------
+\subsection{Conservation of Tracers}
+\label{Apdx_C.5.1}
+
+constraint of conservation of tracers:
+\begin{flalign*}
+&\int\limits_D \nabla \cdot \left( A\;\nabla T \right)\;dv &&&\\
+\\
+&\equiv \sum\limits_{i,j,k}
+ \biggl\{ \biggr.
+ \delta_i
+ \left[
+ A_u^{\,lT} \frac{e_{2u}\,e_{3u}} {e_{1u}} \delta_{i+1/2}
+ \left[ T \right]
+ \right]
+ + \delta_j
+ \left[
+ A_v^{\,lT} \frac{e_{1v}\,e_{3v}} {e_{2v}} \delta_{j+1/2}
+ \left[ T \right]
+ \right]
+ &&\\ & \qquad \qquad \qquad \qquad \qquad \qquad \quad \;\;\;
+ + \delta_k
+ \left[
+ A_w^{\,vT} \frac{e_{1t}\,e_{2t}} {e_{3t}} \delta_{k+1/2}
+ \left[ T \right]
+ \right]
+ \biggr\} \quad \equiv 0
+ &&\\
+\end{flalign*}
+
+In fact, this property simply results from the flux form of the operator.
+
+% -------------------------------------------------------------------------------------------------------------
+% Dissipation of Tracer Variance
+% -------------------------------------------------------------------------------------------------------------
+\subsection{Dissipation of Tracer Variance}
+\label{Apdx_C.5.2}
+
+constraint on the dissipation of tracer variance:
+\begin{flalign*}
+\int\limits_D T\;\nabla & \cdot \left( A\;\nabla T \right)\;dv &&&\\
+&\equiv \sum\limits_{i,j,k} \; T
+\biggl\{ \biggr.
+ \delta_i \left[ A_u^{\,lT} \frac{e_{2u}\,e_{3u}} {e_{1u}} \delta_{i+1/2} \left[T\right] \right]
+& + \delta_j \left[ A_v^{\,lT} \frac{e_{1v} \,e_{3v}} {e_{2v}} \delta_{j+1/2} \left[T\right] \right]
+ \quad&& \\
+ \biggl.
+&&+ \delta_k \left[A_w^{\,vT}\frac{e_{1t}\,e_{2t}} {e_{3t}}\delta_{k+1/2}\left[T\right]\right]
+\biggr\} &&
+\end{flalign*}
+\begin{flalign*}
+\equiv - \sum\limits_{i,j,k}
+ \biggl\{ \biggr. \quad
+ & A_u^{\,lT} \left( \frac{1} {e_{1u}} \delta_{i+1/2} \left[ T \right] \right)^2 e_{1u}\,e_{2u}\,e_{3u} && \\
+ & + A_v^{\,lT} \left( \frac{1} {e_{2v}} \delta_{j+1/2} \left[ T \right] \right)^2 e_{1v}\,e_{2v}\,e_{3v} && \\ \biggl.
+ & + A_w^{\,vT} \left( \frac{1} {e_{3w}} \delta_{k+1/2} \left[ T \right] \right)^2 e_{1w}\,e_{2w}\,e_{3w} \biggr\}
+ \quad \leq 0 && \\
+\end{flalign*}
+
+
+%%%% end of appendix in gm comment
+%}
Index: /branches/UKMO/dev_r5518_GO6_under_ice_relax_dr_hook/DOC/TexFiles/Chapters/Annex_D.tex
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--- /branches/UKMO/dev_r5518_GO6_under_ice_relax_dr_hook/DOC/TexFiles/Chapters/Annex_D.tex (revision 11643)
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+% ================================================================
+% Appendix D Ñ Coding Rules
+% ================================================================
+\chapter{Coding Rules}
+\label{Apdx_D}
+\minitoc
+
+\newpage
+$\ $\newline % force a new ligne
+$\ $\newline % force a new ligne
+
+
+A "model life" is more than ten years. Its software, composed of a few
+hundred modules, is used by many people who are scientists or students
+and do not necessarily know every aspect of computing very well.
+Moreover, a well thought-out program is easier to read and understand,
+less difficult to modify, produces fewer bugs and is easier to maintain.
+Therefore, it is essential that the model development follows some rules :
+
+- well planned and designed
+
+- well written
+
+- well documented (both on- and off-line)
+
+- maintainable
+
+- easily portable
+
+- flexible.
+
+To satisfy part of these aims, \NEMO is written with a coding standard which
+is close to the ECMWF rules, named DOCTOR \citep{Gibson_TR86}.
+These rules present some advantages like :
+
+- to provide a well presented program
+
+- to use rules for variable names which allow recognition of their type
+(integer, real, parameter, local or shared variables, etc. ).
+
+This facilitates both the understanding and the debugging of an algorithm.
+
+% ================================================================
+% The program structure
+% ================================================================
+\section{The program structure}
+\label{Apdx_D_structure}
+
+Each program begins with a set of headline comments containing :
+
+- the program title
+
+- the purpose of the routine
+
+- the method and algorithms used
+
+- the detail of input and output interfaces
+
+- the external routines and functions used (if they exist)
+
+- references (if they exist)
+
+- the author name(s), the date of creation and any updates.
+
+- Each program is split into several well separated sections and
+sub-sections with an underlined title and specific labelled statements.
+
+- A program has not more than 200 to 300 lines.
+
+A template of a module style can be found on the NEMO depository
+in the following file : NEMO/OPA\_SRC/module\_example.
+% ================================================================
+% Coding conventions
+% ================================================================
+\section{Coding conventions}
+\label{Apdx_D_coding}
+
+- Use of the universal language \textsc{Fortran} 90, and try to avoid obsolescent
+features like statement functions, do not use GO TO and EQUIVALENCE statements.
+
+- A continuation line begins with the character {\&} indented by three spaces
+compared to the previous line, while the previous line ended with the character {\&}.
+
+- All the variables must be declared. The code is usually compiled with implicit none.
+
+- Never use continuation lines in the declaration of a variable. When searching a
+variable in the code through a \textit{grep} command, the declaration line will be found.
+
+- In the declaration of a PUBLIC variable, the comment part at the end of the line
+should start with the two characters "\verb?!:?". the following UNIX command, \\
+\verb?grep var_name *90 \ grep \!: ? \\
+will display the module name and the line where the var\_name declaration is.
+
+- Always use a three spaces indentation in DO loop, CASE, or IF-ELSEIF-ELSE-ENDIF
+statements.
+
+- use a space after a comma, except when it appears to separate the indices of an array.
+
+- use call to ctl\_stop routine instead of just a STOP.
+
+
+\newpage
+% ================================================================
+% Naming Conventions
+% ================================================================
+\section{Naming Conventions}
+\label{Apdx_D_naming}
+
+The purpose of the naming conventions is to use prefix letters to classify
+model variables. These conventions allow the variable type to be easily
+known and rapidly identified. The naming conventions are summarised
+in the Table below:
+
+
+%--------------------------------------------------TABLE--------------------------------------------------
+\begin{table}[htbp] \label{Tab_VarName}
+\begin{center}
+\begin{tabular}{|p{45pt}|p{35pt}|p{45pt}|p{40pt}|p{40pt}|p{40pt}|p{40pt}|p{40pt}|}
+\hline Type \par / Status & integer& real& logical & character & structure & double \par precision& complex \\
+\hline
+public \par or \par module variable&
+\textbf{m n} \par \textit{but not} \par \textbf{nn\_}&
+\textbf{a b e f g h o q r} \par \textbf{t} \textit{to} \textbf{x} \par but not \par \textbf{fs rn\_}&
+\textbf{l} \par \textit{but not} \par \textbf{lp ld} \par \textbf{ ll ln\_}&
+\textbf{c} \par \textit{but not} \par \textbf{cp cd} \par \textbf{cl cn\_}&
+\textbf{s} \par \textit{but not} \par \textbf{sd sd} \par \textbf{sl sn\_}&
+\textbf{d} \par \textit{but not} \par \textbf{dp dd} \par \textbf{dl dn\_}&
+\textbf{y} \par \textit{but not} \par \textbf{yp yd} \par \textbf{yl yn} \\
+\hline
+dummy \par argument&
+\textbf{k} \par \textit{but not} \par \textbf{kf}&
+\textbf{p} \par \textit{but not} \par \textbf{pp pf}&
+\textbf{ld}&
+\textbf{cd}&
+\textbf{sd}&
+\textbf{dd}&
+\textbf{yd} \\
+\hline
+local \par variable&
+\textbf{i}&
+\textbf{z}&
+\textbf{ll}&
+\textbf{cl}&
+\textbf{sl}&
+\textbf{dl}&
+\textbf{yl} \\
+\hline
+loop \par control&
+\textbf{j} \par \textit{but not} \par \textbf{jp}&
+&
+&
+&
+&
+&
+ \\
+\hline
+parameter&
+\textbf{jp}&
+\textbf{pp}&
+\textbf{lp}&
+\textbf{cp}&
+\textbf{sp}&
+\textbf{dp}&
+\textbf{yp} \\
+\hline
+
+namelist&
+\textbf{nn\_}&
+\textbf{rn\_}&
+\textbf{ln\_}&
+\textbf{cn\_}&
+\textbf{sn\_}&
+\textbf{dn\_}&
+\textbf{yn\_}
+\\
+\hline
+CPP \par macro&
+\textbf{kf}&
+\textbf{fs} \par &
+&
+&
+&
+&
+ \\
+\hline
+\end{tabular}
+\label{tab1}
+\end{center}
+\end{table}
+%--------------------------------------------------------------------------------------------------------------
+
+\newpage
+% ================================================================
+% The program structure
+% ================================================================
+\section{The program structure}
+\label{Apdx_D_structure}
+
+To be done....
Index: /branches/UKMO/dev_r5518_GO6_under_ice_relax_dr_hook/DOC/TexFiles/Chapters/Annex_E.tex
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--- /branches/UKMO/dev_r5518_GO6_under_ice_relax_dr_hook/DOC/TexFiles/Chapters/Annex_E.tex (revision 11643)
+++ /branches/UKMO/dev_r5518_GO6_under_ice_relax_dr_hook/DOC/TexFiles/Chapters/Annex_E.tex (revision 11643)
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+% ================================================================
+% Appendix E : Note on some algorithms
+% ================================================================
+\chapter{Note on some algorithms}
+\label{Apdx_E}
+\minitoc
+
+\newpage
+$\ $\newline % force a new ligne
+
+ This appendix some on going consideration on algorithms used or planned to be used
+in \NEMO.
+
+$\ $\newline % force a new ligne
+
+% -------------------------------------------------------------------------------------------------------------
+% UBS scheme
+% -------------------------------------------------------------------------------------------------------------
+\section{Upstream Biased Scheme (UBS) (\np{ln\_traadv\_ubs}=T)}
+\label{TRA_adv_ubs}
+
+The UBS advection scheme is an upstream biased third order scheme based on
+an upstream-biased parabolic interpolation. It is also known as Cell Averaged
+QUICK scheme (Quadratic Upstream Interpolation for Convective
+Kinematics). For example, in the $i$-direction :
+\begin{equation} \label{Eq_tra_adv_ubs2}
+\tau _u^{ubs} = \left\{ \begin{aligned}
+ & \tau _u^{cen4} + \frac{1}{12} \,\tau"_i & \quad \text{if }\ u_{i+1/2} \geqslant 0 \\
+ & \tau _u^{cen4} - \frac{1}{12} \,\tau"_{i+1} & \quad \text{if }\ u_{i+1/2} < 0
+ \end{aligned} \right.
+\end{equation}
+or equivalently, the advective flux is
+\begin{equation} \label{Eq_tra_adv_ubs2}
+U_{i+1/2} \ \tau _u^{ubs}
+=U_{i+1/2} \ \overline{ T_i - \frac{1}{6}\,\tau"_i }^{\,i+1/2}
+- \frac{1}{2}\, |U|_{i+1/2} \;\frac{1}{6} \;\delta_{i+1/2}[\tau"_i]
+\end{equation}
+where $U_{i+1/2} = e_{1u}\,e_{3u}\,u_{i+1/2}$ and
+$\tau "_i =\delta _i \left[ {\delta _{i+1/2} \left[ \tau \right]} \right]$.
+By choosing this expression for $\tau "$ we consider a fourth order approximation
+of $\partial_i^2$ with a constant i-grid spacing ($\Delta i=1$).
+
+Alternative choice: introduce the scale factors:
+$\tau "_i =\frac{e_{1T}}{e_{2T}\,e_{3T}}\delta _i \left[ \frac{e_{2u} e_{3u} }{e_{1u} }\delta _{i+1/2}[\tau] \right]$.
+
+
+This results in a dissipatively dominant (i.e. hyper-diffusive) truncation
+error \citep{Shchepetkin_McWilliams_OM05}. The overall performance of the
+advection scheme is similar to that reported in \cite{Farrow1995}.
+It is a relatively good compromise between accuracy and smoothness. It is
+not a \emph{positive} scheme meaning false extrema are permitted but the
+amplitude of such are significantly reduced over the centred second order
+method. Nevertheless it is not recommended to apply it to a passive tracer
+that requires positivity.
+
+The intrinsic diffusion of UBS makes its use risky in the vertical direction
+where the control of artificial diapycnal fluxes is of paramount importance.
+It has therefore been preferred to evaluate the vertical flux using the TVD
+scheme when \np{ln\_traadv\_ubs}=T.
+
+For stability reasons, in \eqref{Eq_tra_adv_ubs}, the first term which corresponds
+to a second order centred scheme is evaluated using the \textit{now} velocity
+(centred in time) while the second term which is the diffusive part of the scheme,
+is evaluated using the \textit{before} velocity (forward in time. This is discussed
+by \citet{Webb_al_JAOT98} in the context of the Quick advection scheme. UBS and QUICK
+schemes only differ by one coefficient. Substituting 1/6 with 1/8 in
+(\ref{Eq_tra_adv_ubs}) leads to the QUICK advection scheme \citep{Webb_al_JAOT98}.
+This option is not available through a namelist parameter, since the 1/6
+coefficient is hard coded. Nevertheless it is quite easy to make the
+substitution in \mdl{traadv\_ubs} module and obtain a QUICK scheme
+
+NB 1: When a high vertical resolution $O(1m)$ is used, the model stability can
+be controlled by vertical advection (not vertical diffusion which is usually
+solved using an implicit scheme). Computer time can be saved by using a
+time-splitting technique on vertical advection. This possibility have been
+implemented and validated in ORCA05-L301. It is not currently offered in the
+current reference version.
+
+NB 2 : In a forthcoming release four options will be proposed for the
+vertical component used in the UBS scheme. $\tau _w^{ubs}$ will be
+evaluated using either \textit{(a)} a centred $2^{nd}$ order scheme ,
+or \textit{(b)} a TVD scheme, or \textit{(c)} an interpolation based on conservative
+parabolic splines following \citet{Shchepetkin_McWilliams_OM05} implementation of UBS in ROMS,
+or \textit{(d)} an UBS. The $3^{rd}$ case has dispersion properties similar to an
+eight-order accurate conventional scheme.
+
+NB 3 : It is straight forward to rewrite \eqref{Eq_tra_adv_ubs} as follows:
+\begin{equation} \label{Eq_tra_adv_ubs2}
+\tau _u^{ubs} = \left\{ \begin{aligned}
+ & \tau _u^{cen4} + \frac{1}{12} \tau"_i & \quad \text{if }\ u_{i+1/2} \geqslant 0 \\
+ & \tau _u^{cen4} - \frac{1}{12} \tau"_{i+1} & \quad \text{if }\ u_{i+1/2} < 0
+ \end{aligned} \right.
+\end{equation}
+or equivalently
+\begin{equation} \label{Eq_tra_adv_ubs2}
+\begin{split}
+e_{2u} e_{3u}\,u_{i+1/2} \ \tau _u^{ubs}
+&= e_{2u} e_{3u}\,u_{i+1/2} \ \overline{ T - \frac{1}{6}\,\tau"_i }^{\,i+1/2} \\
+& - \frac{1}{2} e_{2u} e_{3u}\,|u|_{i+1/2} \;\frac{1}{6} \;\delta_{i+1/2}[\tau"_i]
+\end{split}
+\end{equation}
+\eqref{Eq_tra_adv_ubs2} has several advantages. First it clearly evidence that
+the UBS scheme is based on the fourth order scheme to which is added an
+upstream biased diffusive term. Second, this emphasises that the $4^{th}$ order
+part have to be evaluated at \emph{now} time step, not only the $2^{th}$ order
+part as stated above using \eqref{Eq_tra_adv_ubs}. Third, the diffusive term is
+in fact a biharmonic operator with a eddy coefficient with is simply proportional
+to the velocity.
+
+laplacian diffusion:
+\begin{equation} \label{Eq_tra_ldf_lap}
+\begin{split}
+D_T^{lT} =\frac{1}{e_{1T} \; e_{2T}\; e_{3T} } &\left[ {\quad \delta _i
+\left[ {A_u^{lT} \frac{e_{2u} e_{3u} }{e_{1u} }\;\delta _{i+1/2}
+\left[ T \right]} \right]} \right.
+\\
+&\ \left. {+\; \delta _j \left[
+{A_v^{lT} \left( {\frac{e_{1v} e_{3v} }{e_{2v} }\;\delta _{j+1/2} \left[ T
+\right]} \right)} \right]\quad } \right]
+\end{split}
+\end{equation}
+
+bilaplacian:
+\begin{equation} \label{Eq_tra_ldf_lap}
+\begin{split}
+D_T^{lT} =&-\frac{1}{e_{1T} \; e_{2T}\; e_{3T}} \\
+& \delta _i \left[ \sqrt{A_u^{lT}}\ \frac{e_{2u}\,e_{3u}}{e_{1u}}\;\delta _{i+1/2}
+ \left[ \frac{1}{e_{1T}\,e_{2T}\, e_{3T}}
+ \delta _i \left[ \sqrt{A_u^{lT}}\ \frac{e_{2u}\,e_{3u}}{e_{1u}}\;\delta _{i+1/2}
+ [T] \right] \right] \right]
+\end{split}
+\end{equation}
+with ${A_u^{lT}}^2 = \frac{1}{12} {e_{1u}}^3\ |u|$,
+$i.e.$ $A_u^{lT} = \frac{1}{\sqrt{12}} \,e_{1u}\ \sqrt{ e_{1u}\,|u|\,}$
+it comes :
+\begin{equation} \label{Eq_tra_ldf_lap}
+\begin{split}
+D_T^{lT} =&-\frac{1}{12}\,\frac{1}{e_{1T} \; e_{2T}\; e_{3T}} \\
+& \delta _i \left[ e_{2u}\,e_{3u}\,\sqrt{ e_{1u}\,|u|\,}\;\delta _{i+1/2}
+ \left[ \frac{1}{e_{1T}\,e_{2T}\, e_{3T}}
+ \delta _i \left[ e_{2u}\,e_{3u}\,\sqrt{ e_{1u}\,|u|\,}\;\delta _{i+1/2}
+ [T] \right] \right] \right]
+\end{split}
+\end{equation}
+if the velocity is uniform ($i.e.$ $|u|=cst$) then the diffusive flux is
+\begin{equation} \label{Eq_tra_ldf_lap}
+\begin{split}
+F_u^{lT} = - \frac{1}{12}
+ e_{2u}\,e_{3u}\,|u| \;\sqrt{ e_{1u}}\,\delta _{i+1/2}
+ \left[ \frac{1}{e_{1T}\,e_{2T}\, e_{3T}}
+ \delta _i \left[ e_{2u}\,e_{3u}\,\sqrt{ e_{1u}}\:\delta _{i+1/2}
+ [T] \right] \right]
+\end{split}
+\end{equation}
+beurk.... reverte the logic: starting from the diffusive part of the advective flux it comes:
+
+\begin{equation} \label{Eq_tra_adv_ubs2}
+\begin{split}
+F_u^{lT}
+&= - \frac{1}{2} e_{2u} e_{3u}\,|u|_{i+1/2} \;\frac{1}{6} \;\delta_{i+1/2}[\tau"_i]
+\end{split}
+\end{equation}
+if the velocity is uniform ($i.e.$ $|u|=cst$) and choosing $\tau "_i =\frac{e_{1T}}{e_{2T}\,e_{3T}}\delta _i \left[ \frac{e_{2u} e_{3u} }{e_{1u} } \delta _{i+1/2}[\tau] \right]$
+
+sol 1 coefficient at T-point ( add $e_{1u}$ and $e_{1T}$ on both side of first $\delta$):
+\begin{equation} \label{Eq_tra_adv_ubs2}
+\begin{split}
+F_u^{lT}
+&= - \frac{1}{12} \frac{e_{2u} e_{3u}}{e_{1u}}\;\delta_{i+1/2}\left[ \frac{e_{1T}^3\,|u|}{e_{1T}e_{2T}\,e_{3T}}\,\delta _i \left[ \frac{e_{2u} e_{3u} }{e_{1u} } \delta _{i+1/2}[\tau] \right] \right]
+\end{split}
+\end{equation}
+which leads to ${A_T^{lT}}^2 = \frac{1}{12} {e_{1T}}^3\ \overline{|u|}^{\,i+1/2}$
+
+sol 2 coefficient at u-point: split $|u|$ into $\sqrt{|u|}$ and $e_{1T}$ into $\sqrt{e_{1u}}$
+\begin{equation} \label{Eq_tra_adv_ubs2}
+\begin{split}
+F_u^{lT}
+&= - \frac{1}{12} {e_{1u}}^1 \sqrt{e_{1u}|u|} \frac{e_{2u} e_{3u}}{e_{1u}}\;\delta_{i+1/2}\left[ \frac{1}{e_{2T}\,e_{3T}}\,\delta _i \left[ \sqrt{e_{1u}|u|} \frac{e_{2u} e_{3u} }{e_{1u} } \delta _{i+1/2}[\tau] \right] \right] \\
+&= - \frac{1}{12} e_{1u} \sqrt{e_{1u}|u|\,} \frac{e_{2u} e_{3u}}{e_{1u}}\;\delta_{i+1/2}\left[ \frac{1}{e_{1T}\,e_{2T}\,e_{3T}}\,\delta _i \left[ e_{1u} \sqrt{e_{1u}|u|\,} \frac{e_{2u} e_{3u} }{e_{1u}} \delta _{i+1/2}[\tau] \right] \right]
+\end{split}
+\end{equation}
+which leads to ${A_u^{lT}} = \frac{1}{12} {e_{1u}}^3\ |u|$
+
+
+% -------------------------------------------------------------------------------------------------------------
+% Leap-Frog energetic
+% -------------------------------------------------------------------------------------------------------------
+\section{Leap-Frog energetic }
+\label{LF}
+
+We adopt the following semi-discrete notation for time derivative. Given the values of a variable $q$ at successive time step, the time derivation and averaging operators at the mid time step are:
+\begin{subequations} \label{dt_mt}
+\begin{align}
+ \delta _{t+\rdt/2} [q] &= \ \ \, q^{t+\rdt} - q^{t} \\
+ \overline q^{\,t+\rdt/2} &= \left\{ q^{t+\rdt} + q^{t} \right\} \; / \; 2
+\end{align}
+\end{subequations}
+As for space operator, the adjoint of the derivation and averaging time operators are
+$\delta_t^*=\delta_{t+\rdt/2}$ and $\overline{\cdot}^{\,t\,*}= \overline{\cdot}^{\,t+\Delta/2}$
+, respectively.
+
+The Leap-frog time stepping given by \eqref{Eq_DOM_nxt} can be defined as:
+\begin{equation} \label{LF}
+ \frac{\partial q}{\partial t}
+ \equiv \frac{1}{\rdt} \overline{ \delta _{t+\rdt/2}[q]}^{\,t}
+ = \frac{q^{t+\rdt}-q^{t-\rdt}}{2\rdt}
+\end{equation}
+Note that \eqref{LF} shows that the leapfrog time step is $\rdt$, not $2\rdt$
+as it can be found sometime in literature.
+The leap-Frog time stepping is a second order centered scheme. As such it respects
+the quadratic invariant in integral forms, $i.e.$ the following continuous property,
+\begin{equation} \label{Energy}
+\int_{t_0}^{t_1} {q\, \frac{\partial q}{\partial t} \;dt}
+ =\int_{t_0}^{t_1} {\frac{1}{2}\, \frac{\partial q^2}{\partial t} \;dt}
+ = \frac{1}{2} \left( {q_{t_1}}^2 - {q_{t_0}}^2 \right) ,
+\end{equation}
+is satisfied in discrete form. Indeed,
+\begin{equation} \begin{split}
+\int_{t_0}^{t_1} {q\, \frac{\partial q}{\partial t} \;dt}
+ &\equiv \sum\limits_{0}^{N}
+ {\frac{1}{\rdt} q^t \ \overline{ \delta _{t+\rdt/2}[q]}^{\,t} \ \rdt}
+ \equiv \sum\limits_{0}^{N} { q^t \ \overline{ \delta _{t+\rdt/2}[q]}^{\,t} } \\
+ &\equiv \sum\limits_{0}^{N} { \overline{q}^{\,t+\Delta/2}{ \delta _{t+\rdt/2}[q]}}
+ \equiv \sum\limits_{0}^{N} { \frac{1}{2} \delta _{t+\rdt/2}[q^2] }\\
+ &\equiv \sum\limits_{0}^{N} { \frac{1}{2} \delta _{t+\rdt/2}[q^2] }
+ \equiv \frac{1}{2} \left( {q_{t_1}}^2 - {q_{t_0}}^2 \right)
+\end{split} \end{equation}
+NB here pb of boundary condition when applying the adjoin! In space, setting to 0
+the quantity in land area is sufficient to get rid of the boundary condition
+(equivalently of the boundary value of the integration by part). In time this boundary
+condition is not physical and \textbf{add something here!!!}
+
+
+
+
+
+
+% ================================================================
+% Iso-neutral diffusion :
+% ================================================================
+
+\section{Lateral diffusion operator}
+
+% ================================================================
+% Griffies' iso-neutral diffusion operator :
+% ================================================================
+\subsection{Griffies' iso-neutral diffusion operator}
+
+Let try to define a scheme that get its inspiration from the \citet{Griffies_al_JPO98}
+scheme, but is formulated within the \NEMO framework ($i.e.$ using scale
+factors rather than grid-size and having a position of $T$-points that is not
+necessary in the middle of vertical velocity points, see Fig.~\ref{Fig_zgr_e3}).
+
+In the formulation \eqref{Eq_tra_ldf_iso} introduced in 1995 in OPA, the ancestor of \NEMO,
+the off-diagonal terms of the small angle diffusion tensor contain several double
+spatial averages of a gradient, for example $\overline{\overline{\delta_k \cdot}}^{\,i,k}$.
+It is apparent that the combination of a $k$ average and a $k$ derivative of the tracer
+allows for the presence of grid point oscillation structures that will be invisible
+to the operator. These structures are \textit{computational modes}. They
+will not be damped by the iso-neutral operator, and even possibly amplified by it.
+In other word, the operator applied to a tracer does not warranties the decrease of
+its global average variance. To circumvent this, we have introduced a smoothing of
+the slopes of the iso-neutral surfaces (see \S\ref{LDF}). Nevertheless, this technique
+works fine for $T$ and $S$ as they are active tracers ($i.e.$ they enter the computation
+of density), but it does not work for a passive tracer. \citep{Griffies_al_JPO98} introduce
+a different way to discretise the off-diagonal terms that nicely solve the problem.
+The idea is to get rid of combinations of an averaged in one direction combined
+with a derivative in the same direction by considering triads. For example in the
+(\textbf{i},\textbf{k}) plane, the four triads are defined at the $(i,k)$ $T$-point as follows:
+\begin{equation} \label{Gf_triads}
+_i^k \mathbb{T}_{i_p}^{k_p} (T)
+= \frac{1}{4} \ {b_u}_{\,i+i_p}^{\,k} \ A_i^k \left(
+ \frac{ \delta_{i + i_p}[T^k] }{ {e_{1u}}_{\,i + i_p}^{\,k} }
+-\ {_i^k \mathbb{R}_{i_p}^{k_p}} \ \frac{ \delta_{k+k_p} [T^i] }{ {e_{3w}}_{\,i}^{\,k+k_p} }
+ \right)
+\end{equation}
+where the indices $i_p$ and $k_p$ define the four triads and take the following value:
+$i_p = -1/2$ or $1/2$ and $k_p = -1/2$ or $1/2$,
+$b_u= e_{1u}\,e_{2u}\,e_{3u}$ is the volume of $u$-cells,
+$A_i^k$ is the lateral eddy diffusivity coefficient defined at $T$-point,
+and $_i^k \mathbb{R}_{i_p}^{k_p}$ is the slope associated with each triad :
+\begin{equation} \label{Gf_slopes}
+_i^k \mathbb{R}_{i_p}^{k_p}
+=\frac{ {e_{3w}}_{\,i}^{\,k+k_p}} { {e_{1u}}_{\,i+i_p}^{\,k}} \ \frac
+{\left(\alpha / \beta \right)_i^k \ \delta_{i + i_p}[T^k] - \delta_{i + i_p}[S^k] }
+{\left(\alpha / \beta \right)_i^k \ \delta_{k+k_p}[T^i ] - \delta_{k+k_p}[S^i ] }
+\end{equation}
+Note that in \eqref{Gf_slopes} we use the ratio $\alpha / \beta$ instead of
+multiplying the temperature derivative by $\alpha$ and the salinity derivative
+by $\beta$. This is more efficient as the ratio $\alpha / \beta$ can to be
+evaluated directly.
+
+Note that in \eqref{Gf_triads}, we chose to use ${b_u}_{\,i+i_p}^{\,k}$ instead of
+${b_{uw}}_{\,i+i_p}^{\,k+k_p}$. This choice has been motivated by the decrease
+of tracer variance and the presence of partial cell at the ocean bottom
+(see Appendix~\ref{Apdx_Gf_operator}).
+
+%>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>
+\begin{figure}[!ht] \label{Fig_ISO_triad}
+\begin{center}
+\includegraphics[width=0.70\textwidth]{./TexFiles/Figures/Fig_ISO_triad.pdf}
+\caption{ \label{Fig_ISO_triad}
+Triads used in the Griffies's like iso-neutral diffision scheme for
+$u$-component (upper panel) and $w$-component (lower panel).}
+\end{center}
+\end{figure}
+%>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>
+
+The four iso-neutral fluxes associated with the triads are defined at $T$-point.
+They take the following expression :
+\begin{flalign} \label{Gf_fluxes}
+\begin{split}
+{_i^k {\mathbb{F}_u}_{i_p}^{k_p} } (T)
+ &= \ \; \qquad \quad { _i^k \mathbb{T}_{i_p}^{k_p} }(T) \;\ / \ { {e_{1u}}_{\,i+i_p}^{\,k}} \\
+{_i^k {\mathbb{F}_w}_{i_p}^{k_p} } (T)
+ &= -\; { _i^k \mathbb{R}_{i_p}^{k_p} }
+ \ \; { _i^k \mathbb{T}_{i_p}^{k_p} }(T) \;\ / \ { {e_{3w}}_{\,i}^{\,k+k_p}}
+\end{split}
+\end{flalign}
+
+The resulting iso-neutral fluxes at $u$- and $w$-points are then given by the
+sum of the fluxes that cross the $u$- and $w$-face (Fig.~\ref{Fig_ISO_triad}):
+\begin{flalign} \label{Eq_iso_flux}
+\textbf{F}_{iso}(T)
+&\equiv \sum_{\substack{i_p,\,k_p}}
+ \begin{pmatrix}
+ {_{i+1/2-i_p}^k {\mathbb{F}_u}_{i_p}^{k_p} } (T) \\
+ \\
+ {_i^{k+1/2-k_p} {\mathbb{F}_w}_{i_p}^{k_p} } (T) \\
+ \end{pmatrix} \notag \\
+& \notag \\
+&\equiv \sum_{\substack{i_p,\,k_p}}
+ \begin{pmatrix}
+ && { _{i+1/2-i_p}^k \mathbb{T}_{i_p}^{k_p} }(T) \;\ / \ { {e_{1u}}_{\,i+1/2}^{\,k} } \\
+ \\
+ & -\; { _i^{k+1/2-k_p} \mathbb{R}_{i_p}^{k_p} }
+ & {_i^{k+1/2-k_p} \mathbb{T}_{i_p}^{k_p} }(T) \;\ / \ { {e_{3w}}_{\,i}^{\,k+1/2} } \\
+ \end{pmatrix} % \\
+% &\\
+% &\equiv \sum_{\substack{i_p,\,k_p}}
+% \begin{pmatrix}
+% \qquad \qquad \qquad
+% \frac{1}{ {e_{1u}}_{\,i+1/2}^{\,k} } \ \;
+% { _{i+1/2-i_p}^k \mathbb{T}_{i_p}^{k_p} }(T)\\
+% \\
+% -\frac{1}{ {e_{3w}}_{\,i}^{\,k+1/2} } \ \;
+% { _i^{k+1/2-k_p} \mathbb{R}_{i_p}^{k_p} } \ \;
+% {_i^{k+1/2-k_p} \mathbb{T}_{i_p}^{k_p} }(T)\\
+% \end{pmatrix}
+\end{flalign}
+resulting in a iso-neutral diffusion tendency on temperature given by the divergence
+of the sum of all the four triad fluxes :
+\begin{equation} \label{Gf_operator}
+D_l^T = \frac{1}{b_T} \sum_{\substack{i_p,\,k_p}} \left\{
+ \delta_{i} \left[{_{i+1/2-i_p}^k {\mathbb{F}_u }_{i_p}^{k_p}} \right]
+ + \delta_{k} \left[ {_i^{k+1/2-k_p} {\mathbb{F}_w}_{i_p}^{k_p}} \right] \right\}
+\end{equation}
+where $b_T= e_{1T}\,e_{2T}\,e_{3T}$ is the volume of $T$-cells.
+
+This expression of the iso-neutral diffusion has been chosen in order to satisfy
+the following six properties:
+\begin{description}
+\item[$\bullet$ horizontal diffusion] The discretization of the diffusion operator
+recovers the traditional five-point Laplacian in the limit of flat iso-neutral direction :
+\begin{equation} \label{Gf_property1a}
+D_l^T = \frac{1}{b_T} \ \delta_{i}
+ \left[ \frac{e_{2u}\,e_{3u}}{e_{1u}} \; \overline{A}^{\,i} \; \delta_{i+1/2}[T] \right]
+\qquad \text{when} \quad
+ { _i^k \mathbb{R}_{i_p}^{k_p} }=0
+\end{equation}
+
+\item[$\bullet$ implicit treatment in the vertical] In the diagonal term associated
+with the vertical divergence of the iso-neutral fluxes (i.e. the term associated
+with a second order vertical derivative) appears only tracer values associated
+with a single water column. This is of paramount importance since it means
+that the implicit in time algorithm for solving the vertical diffusion equation can
+be used to evaluate this term. It is a necessity since the vertical eddy diffusivity
+associated with this term,
+\begin{equation}
+ \sum_{\substack{i_p, \,k_p}} \left\{
+ A_i^k \; \left(_i^k \mathbb{R}_{i_p}^{k_p}\right)^2
+ \right\}
+\end{equation}
+can be quite large.
+
+\item[$\bullet$ pure iso-neutral operator] The iso-neutral flux of locally referenced
+potential density is zero, $i.e.$
+\begin{align} \label{Gf_property2}
+\begin{matrix}
+&{_i^k {\mathbb{F}_u}_{i_p}^{k_p} (\rho)}
+ &= &\alpha_i^k &{_i^k {\mathbb{F}_u}_{i_p}^{k_p} } (T)
+ &- \ \; \beta _i^k &{_i^k {\mathbb{F}_u}_{i_p}^{k_p} } (S) & = \ 0 \\
+&{_i^k {\mathbb{F}_w}_{i_p}^{k_p} (\rho)}
+ &= &\alpha_i^k &{_i^k {\mathbb{F}_w}_{i_p}^{k_p} } (T)
+ &- \ \; \beta _i^k &{_i^k {\mathbb{F}_w}_{i_p}^{k_p} } (S) &= \ 0
+\end{matrix}
+\end{align}
+This result is trivially obtained using the \eqref{Gf_triads} applied to $T$ and $S$
+and the definition of the triads' slopes \eqref{Gf_slopes}.
+
+\item[$\bullet$ conservation of tracer] The iso-neutral diffusion term conserve the
+total tracer content, $i.e.$
+\begin{equation} \label{Gf_property1}
+\sum_{i,j,k} \left\{ D_l^T \ b_T \right\} = 0
+\end{equation}
+This property is trivially satisfied since the iso-neutral diffusive operator
+is written in flux form.
+
+\item[$\bullet$ decrease of tracer variance] The iso-neutral diffusion term does
+not increase the total tracer variance, $i.e.$
+\begin{equation} \label{Gf_property1}
+\sum_{i,j,k} \left\{ T \ D_l^T \ b_T \right\} \leq 0
+\end{equation}
+The property is demonstrated in the Appendix~\ref{Apdx_Gf_operator}. It is a
+key property for a diffusion term. It means that the operator is also a dissipation
+term, $i.e.$ it is a sink term for the square of the quantity on which it is applied.
+It therfore ensure that, when the diffusivity coefficient is large enough, the field
+on which it is applied become free of grid-point noise.
+
+\item[$\bullet$ self-adjoint operator] The iso-neutral diffusion operator is self-adjoint,
+$i.e.$
+\begin{equation} \label{Gf_property1}
+\sum_{i,j,k} \left\{ S \ D_l^T \ b_T \right\} = \sum_{i,j,k} \left\{ D_l^S \ T \ b_T \right\}
+\end{equation}
+In other word, there is no needs to develop a specific routine from the adjoint of this
+operator. We just have to apply the same routine. This properties can be demonstrated
+quite easily in a similar way the "non increase of tracer variance" property has been
+proved (see Appendix~\ref{Apdx_Gf_operator}).
+\end{description}
+
+
+$\ $\newline %force an empty line
+% ================================================================
+% Skew flux formulation for Eddy Induced Velocity :
+% ================================================================
+\subsection{ Eddy induced velocity and Skew flux formulation}
+
+When Gent and McWilliams [1990] diffusion is used (\key{traldf\_eiv} defined),
+an additional advection term is added. The associated velocity is the so called
+eddy induced velocity, the formulation of which depends on the slopes of iso-
+neutral surfaces. Contrary to the case of iso-neutral mixing, the slopes used
+here are referenced to the geopotential surfaces, $i.e.$ \eqref{Eq_ldfslp_geo}
+is used in $z$-coordinate, and the sum \eqref{Eq_ldfslp_geo}
++ \eqref{Eq_ldfslp_iso} in $z^*$ or $s$-coordinates.
+
+The eddy induced velocity is given by:
+\begin{equation} \label{Eq_eiv_v}
+\begin{split}
+ u^* & = - \frac{1}{e_2\,e_{3}} \;\partial_k \left( e_2 \, A_e \; r_i \right)
+ = - \frac{1}{e_3} \;\partial_k \left( A_e \; r_i \right) \\
+ v^* & = - \frac{1}{e_1\,e_3}\; \partial_k \left( e_1 \, A_e \; r_j \right)
+ = - \frac{1}{e_3} \;\partial_k \left( A_e \; r_j \right) \\
+w^* & = \frac{1}{e_1\,e_2}\; \left\{ \partial_i \left( e_2 \, A_e \; r_i \right)
+ + \partial_j \left( e_1 \, A_e \;r_j \right) \right\} \\
+\end{split}
+\end{equation}
+where $A_{e}$ is the eddy induced velocity coefficient, and $r_i$ and $r_j$ the
+slopes between the iso-neutral and the geopotential surfaces.
+%%gm wrong: to be modified with 2 2D streamfunctions
+ In other words,
+the eddy induced velocity can be derived from a vector streamfuntion, $\phi$, which
+is given by $\phi = A_e\,\textbf{r}$ as $\textbf{U}^* = \textbf{k} \times \nabla \phi$
+%%end gm
+
+A traditional way to implement this additional advection is to add it to the eulerian
+velocity prior to compute the tracer advection. This allows us to take advantage of
+all the advection schemes offered for the tracers (see \S\ref{TRA_adv}) and not just
+a $2^{nd}$ order advection scheme. This is particularly useful for passive tracers
+where \emph{positivity} of the advection scheme is of paramount importance.
+% give here the expression using the triads. It is different from the one given in \eqref{Eq_ldfeiv}
+% see just below a copy of this equation:
+%\begin{equation} \label{Eq_ldfeiv}
+%\begin{split}
+% u^* & = \frac{1}{e_{2u}e_{3u}}\; \delta_k \left[e_{2u} \, A_{uw}^{eiv} \; \overline{r_{1w}}^{\,i+1/2} \right]\\
+% v^* & = \frac{1}{e_{1u}e_{3v}}\; \delta_k \left[e_{1v} \, A_{vw}^{eiv} \; \overline{r_{2w}}^{\,j+1/2} \right]\\
+%w^* & = \frac{1}{e_{1w}e_{2w}}\; \left\{ \delta_i \left[e_{2u} \, A_{uw}^{eiv} \; \overline{r_{1w}}^{\,i+1/2} \right] + %\delta_j \left[e_{1v} \, A_{vw}^{eiv} \; \overline{r_{2w}}^{\,j+1/2} \right] \right\} \\
+%\end{split}
+%\end{equation}
+\begin{equation} \label{Eq_eiv_vd}
+\textbf{F}_{eiv}^T \equiv \left( \begin{aligned}
+ \sum_{\substack{i_p,\,k_p}} &
+ +{e_{2u}}_{i+1/2-i_p}^{k} \ \ {A_{e}}_{i+1/2-i_p}^{k}
+\ \ \ { _{i+1/2-i_p}^k \mathbb{R}_{i_p}^{k_p} } \ \ \delta_{k+k_p}[T_{i+1/2-i_p}] \\
+ \\
+ \sum_{\substack{i_p,\,k_p}} &
+ - {e_{2u}}_i^{k+1/2-k_p} \ {A_{e}}_i^{k+1/2-k_p}
+\ \ { _i^{k+1/2-k_p} \mathbb{R}_{i_p}^{k_p} } \ \delta_{i+i_p}[T^{k+1/2-k_p}] \\
+\end{aligned} \right)
+\end{equation}
+
+\ref{Griffies_JPO98} introduces another way to implement the eddy induced advection,
+the so-called skew form. It is based on a transformation of the advective fluxes
+using the non-divergent nature of the eddy induced velocity.
+For example in the (\textbf{i},\textbf{k}) plane, the tracer advective fluxes can be
+transformed as follows:
+\begin{flalign*}
+\begin{split}
+\textbf{F}_{eiv}^T =
+\begin{pmatrix}
+ {e_{2}\,e_{3}\; u^*} \\
+ {e_{1}\,e_{2}\; w^*} \\
+\end{pmatrix} \; T
+&=
+\begin{pmatrix}
+ { - \partial_k \left( e_{2} \, A_{e} \; r_i \right) \; T \;} \\
+ {+ \partial_i \left( e_{2} \, A_{e} \; r_i \right) \; T \;} \\
+\end{pmatrix} \\
+&=
+\begin{pmatrix}
+ { - \partial_k \left( e_{2} \, A_{e} \; r_i \; T \right) \;} \\
+ {+ \partial_i \left( e_{2} \, A_{e} \; r_i \; T \right) \;} \\
+\end{pmatrix}
+ +
+\begin{pmatrix}
+ {+ e_{2} \, A_{e} \; r_i \; \partial_k T} \\
+ { - e_{2} \, A_{e} \; r_i \; \partial_i T} \\
+\end{pmatrix}
+\end{split}
+\end{flalign*}
+and since the eddy induces velocity field is no-divergent, we end up with the skew
+form of the eddy induced advective fluxes:
+\begin{equation} \label{Eq_eiv_skew_continuous}
+\textbf{F}_{eiv}^T = \begin{pmatrix}
+ {+ e_{2} \, A_{e} \; r_i \; \partial_k T} \\
+ { - e_{2} \, A_{e} \; r_i \; \partial_i T} \\
+ \end{pmatrix}
+\end{equation}
+The tendency associated with eddy induced velocity is then simply the divergence
+of the \eqref{Eq_eiv_skew_continuous} fluxes. It naturally conserves the tracer
+content, as it is expressed in flux form and, as the advective form, it preserve the
+tracer variance. Another interesting property of \eqref{Eq_eiv_skew_continuous}
+form is that when $A=A_e$, a simplification occurs in the sum of the iso-neutral
+diffusion and eddy induced velocity terms:
+\begin{flalign} \label{Eq_eiv_skew+eiv_continuous}
+\textbf{F}_{iso}^T + \textbf{F}_{eiv}^T &=
+\begin{pmatrix}
+ + \frac{e_2\,e_3\,}{e_1} A \;\partial_i T - e_2 \, A \; r_i \;\partial_k T \\
+ - e_2 \, A_{e} \; r_i \;\partial_i T + \frac{e_1\,e_2}{e_3} \, A \; r_i^2 \;\partial_k T \\
+\end{pmatrix}
++
+\begin{pmatrix}
+ {+ e_{2} \, A_{e} \; r_i \; \partial_k T} \\
+ { - e_{2} \, A_{e} \; r_i \; \partial_i T} \\
+\end{pmatrix} \\
+&= \begin{pmatrix}
+ + \frac{e_2\,e_3\,}{e_1} A \;\partial_i T \\
+ - 2\; e_2 \, A_{e} \; r_i \;\partial_i T + \frac{e_1\,e_2}{e_3} \, A \; r_i^2 \;\partial_k T \\
+\end{pmatrix}
+\end{flalign}
+The horizontal component reduces to the one use for an horizontal laplacian
+operator and the vertical one keep the same complexity, but not more. This property
+has been used to reduce the computational time \citep{Griffies_JPO98}, but it is
+not of practical use as usually $A \neq A_e$. Nevertheless this property can be used to
+choose a discret form of \eqref{Eq_eiv_skew_continuous} which is consistent with the
+iso-neutral operator \eqref{Gf_operator}. Using the slopes \eqref{Gf_slopes}
+and defining $A_e$ at $T$-point($i.e.$ as $A$, the eddy diffusivity coefficient),
+the resulting discret form is given by:
+\begin{equation} \label{Eq_eiv_skew}
+\textbf{F}_{eiv}^T \equiv \frac{1}{4} \left( \begin{aligned}
+ \sum_{\substack{i_p,\,k_p}} &
+ +{e_{2u}}_{i+1/2-i_p}^{k} \ \ {A_{e}}_{i+1/2-i_p}^{k}
+\ \ \ { _{i+1/2-i_p}^k \mathbb{R}_{i_p}^{k_p} } \ \ \delta_{k+k_p}[T_{i+1/2-i_p}] \\
+ \\
+ \sum_{\substack{i_p,\,k_p}} &
+ - {e_{2u}}_i^{k+1/2-k_p} \ {A_{e}}_i^{k+1/2-k_p}
+\ \ { _i^{k+1/2-k_p} \mathbb{R}_{i_p}^{k_p} } \ \delta_{i+i_p}[T^{k+1/2-k_p}] \\
+\end{aligned} \right)
+\end{equation}
+Note that \eqref{Eq_eiv_skew} is valid in $z$-coordinate with or without partial cells.
+In $z^*$ or $s$-coordinate, the slope between the level and the geopotential surfaces
+must be added to $\mathbb{R}$ for the discret form to be exact.
+
+Such a choice of discretisation is consistent with the iso-neutral operator as it uses the
+same definition for the slopes. It also ensures the conservation of the tracer variance
+(see Appendix \ref{Apdx_eiv_skew}), $i.e.$ it does not include a diffusive component
+but is a "pure" advection term.
+
+
+
+
+$\ $\newpage %force an empty line
+% ================================================================
+% Discrete Invariants of the iso-neutral diffrusion
+% ================================================================
+\subsection{Discrete Invariants of the iso-neutral diffrusion}
+\label{Apdx_Gf_operator}
+
+Demonstration of the decrease of the tracer variance in the (\textbf{i},\textbf{j}) plane.
+
+This part will be moved in an Appendix.
+
+The continuous property to be demonstrated is :
+\begin{align*}
+\int_D D_l^T \; T \;dv \leq 0
+\end{align*}
+The discrete form of its left hand side is obtained using \eqref{Eq_iso_flux}
+
+\begin{align*}
+&\int_D D_l^T \; T \;dv \equiv \sum_{i,k} \left\{ T \ D_l^T \ b_T \right\} \\
+&\equiv + \sum_{i,k} \sum_{\substack{i_p,\,k_p}} \left\{
+ \delta_{i} \left[{_{i+1/2-i_p}^k {\mathbb{F}_u }_{i_p}^{k_p}} \right]
+ + \delta_{k} \left[ {_i^{k+1/2-k_p} {\mathbb{F}_w}_{i_p}^{k_p}} \right] \ T \right\} \\
+&\equiv - \sum_{i,k} \sum_{\substack{i_p,\,k_p}} \left\{
+ {_{i+1/2-i_p}^k {\mathbb{F}_u }_{i_p}^{k_p}} \ \delta_{i+1/2} [T]
+ + {_i^{k+1/2-k_p} {\mathbb{F}_w}_{i_p}^{k_p}} \ \delta_{k+1/2} [T] \right\} \\
+&\equiv -\sum_{i,k} \sum_{\substack{i_p,\,k_p}} \left\{
+ \frac{ _{i+1/2-i_p}^k \mathbb{T}_{i_p}^{k_p} (T) }{ {e_{1u}}_{\,i+1/2}^{\,k} } \ \delta_{i+1/2} [T]
+ - { _i^{k+1/2-k_p} \mathbb{R}_{i_p}^{k_p} } \ \;
+ \frac{ _i^{k+1/2-k_p} \mathbb{T}_{i_p}^{k_p} (T) }{ {e_{3w}}_{\,i}^{\,k+1/2} } \ \delta_{k+1/2} [T]
+ \right\} \\
+%
+\allowdisplaybreaks
+\intertext{ Expending the summation on $i_p$ and $k_p$, it becomes:}
+%
+&\equiv -\sum_{i,k}
+\begin{Bmatrix}
+&\ \ \Bigl( { _{i+1}^{k} \mathbb{T}_{-1/2}^{-1/2} (T) }
+&\frac{ \delta_{i +1/2} [T] }{{e_{1u} }_{\,i+1/2}^{\,k}}
+& -\ \ {_{i}^{k+1} \mathbb{R}_{-1/2}^{-1/2}}
+& {_{i}^{k+1} \mathbb{T}_{-1/2}^{-1/2} (T) }
+&\frac{ \delta_{k+1/2} [T] }{{e_{3w}}_{\,i}^{\,k+1/2}} \Bigr)
+& \\
+&+\Bigl( \ \;\; { _i^k \mathbb{T}_{+1/2}^{-1/2} (T) }
+&\frac{ \delta_{i +1/2} [T] }{{e_{1u} }_{\,i+1/2}^{\,k}}
+& -\ \ {_i^{k+1} \mathbb{R}_{+1/2}^{-1/2}}
+& { _i^{k+1} \mathbb{T}_{+1/2}^{-1/2} (T) }
+&\frac{ \delta_{k+1/2} [T] }{{e_{3w}}_{\,i}^{\,k+1/2}} \Bigr)
+& \\
+&+\Bigl( { _{i+1}^{k} \mathbb{T}_{-1/2}^{+1/2} (T) }
+&\frac{ \delta_{i +1/2} [T] }{{e_{1u} }_{\,i+1/2}^{\,k}}
+& -\ \ \ \;\;{_{i}^{k} \mathbb{R}_{-1/2}^{+1/2}}
+& \ \;\;{_{i}^{k} \mathbb{T}_{-1/2}^{+1/2} (T) }
+&\frac{ \delta_{k+1/2} [T] }{{e_{3w}}_{\,i}^{\,k+1/2}} \Bigr)
+& \\
+&+\Bigl( \ \;\; { _{i}^{k} \mathbb{T}_{+1/2}^{+1/2} (T) }
+&\frac{ \delta_{i +1/2} [T] }{{e_{1u} }_{\,i+1/2}^{\,k}}
+& -\ \ \ \;\;{_{i}^{k} \mathbb{R}_{+1/2}^{+1/2}}
+& \ \;\;{_{i}^{k} \mathbb{T}_{+1/2}^{+1/2} (T) }
+&\frac{ \delta_{k+1/2} [T] }{{e_{3w}}_{\,i}^{\,k+1/2}} \Bigr) \\
+\end{Bmatrix}
+%
+\allowdisplaybreaks
+\intertext{The summation is done over all $i$ and $k$ indices, it is therefore possible to introduce a shift of $-1$ either in $i$ or $k$ direction in order to regroup all the terms of the summation by triad at a ($i$,$k$) point. In other words, we regroup all the terms in the neighbourhood that contain a triad at the same ($i$,$k$) indices. It becomes: }
+%
+&\equiv -\sum_{i,k}
+\begin{Bmatrix}
+&\ \ \Bigl( {_i^k \mathbb{T}_{-1/2}^{-1/2} (T) }
+&\frac{ \delta_{i -1/2} [T] }{{e_{1u} }_{\,i-1/2}^{\,k}}
+& -\ \ {_i^k \mathbb{R}_{-1/2}^{-1/2}}
+& {_i^k \mathbb{T}_{-1/2}^{-1/2} (T) }
+&\frac{ \delta_{k-1/2} [T] }{{e_{3w}}_{\,i}^{\,k-1/2}} \Bigr)
+& \\
+&+\Bigl( { _i^k \mathbb{T}_{+1/2}^{-1/2} (T) }
+&\frac{ \delta_{i +1/2} [T] }{{e_{1u} }_{\,i+1/2}^{\,k}}
+& -\ \ {_i^k \mathbb{R}_{+1/2}^{-1/2}}
+& { _i^k \mathbb{T}_{+1/2}^{-1/2} (T) }
+&\frac{ \delta_{k-1/2} [T] }{{e_{3w}}_{\,i}^{\,k-1/2}} \Bigr)
+& \\
+&+\Bigl( {_i^k \mathbb{T}_{-1/2}^{+1/2} (T) }
+&\frac{ \delta_{i -1/2} [T] }{{e_{1u} }_{\,i-1/2}^{\,k}}
+& -\ \ {_i^k \mathbb{R}_{-1/2}^{+1/2}}
+& {_i^k \mathbb{T}_{-1/2}^{+1/2} (T) }
+&\frac{ \delta_{k+1/2} [T] }{{e_{3w}}_{\,i}^{\,k+1/2}} \Bigr)
+& \\
+&+\Bigl( { _i^k \mathbb{T}_{+1/2}^{+1/2} (T) }
+&\frac{ \delta_{i +1/2} [T] }{{e_{1u} }_{\,i+1/2}^{\,k}}
+& -\ \ {_i^k \mathbb{R}_{+1/2}^{+1/2}}
+& {_i^k \mathbb{T}_{+1/2}^{+1/2} (T) }
+&\frac{ \delta_{k+1/2} [T] }{{e_{3w}}_{\,i}^{\,k+1/2}} \Bigr) \\
+\end{Bmatrix} \\
+%
+\allowdisplaybreaks
+\intertext{Then outing in factor the triad in each of the four terms of the summation and substituting the triads by their expression given in \eqref{Gf_triads}. It becomes: }
+%
+&\equiv -\sum_{i,k}
+\begin{Bmatrix}
+&\ \ \Bigl( \frac{ \delta_{i -1/2} [T] }{{e_{1u} }_{\,i-1/2}^{\,k}}
+& -\ \ {_i^k \mathbb{R}_{-1/2}^{-1/2}}
+&\frac{ \delta_{k-1/2} [T] }{{e_{3w}}_{\,i}^{\,k-1/2}} \Bigr)^2
+& \frac{1}{4} \ {b_u}_{\,i-1/2}^{\,k} \ A_i^k
+& \\
+&+\Bigl( \frac{ \delta_{i +1/2} [T] }{{e_{1u} }_{\,i+1/2}^{\,k}}
+& -\ \ {_i^k \mathbb{R}_{+1/2}^{-1/2}}
+&\frac{ \delta_{k-1/2} [T] }{{e_{3w}}_{\,i}^{\,k-1/2}} \Bigr)^2
+& \frac{1}{4} \ {b_u}_{\,i+1/2}^{\,k} \ A_i^k
+& \\
+&+\Bigl( \frac{ \delta_{i -1/2} [T] }{{e_{1u} }_{\,i-1/2}^{\,k}}
+& -\ \ {_i^k \mathbb{R}_{-1/2}^{+1/2}}
+&\frac{ \delta_{k+1/2} [T] }{{e_{3w}}_{\,i}^{\,k+1/2}} \Bigr)^2
+& \frac{1}{4} \ {b_u}_{\,i-1/2}^{\,k} \ A_i^k
+& \\
+&+\Bigl( \frac{ \delta_{i +1/2} [T] }{{e_{1u} }_{\,i+1/2}^{\,k}}
+& -\ \ {_i^k \mathbb{R}_{+1/2}^{+1/2}}
+&\frac{ \delta_{k+1/2} [T] }{{e_{3w}}_{\,i}^{\,k+1/2}} \Bigr)^2
+& \frac{1}{4} \ {b_u}_{\,i+1/2}^{\,k} \ A_i^k \\
+\end{Bmatrix} \\
+& \\
+%
+&\equiv - \sum_{i,k} \sum_{\substack{i_p,\,k_p}} \left\{
+\begin{matrix}
+&\Bigl( \frac{ \delta_{i +i_p} [T] }{{e_{1u} }_{\,i+i_p}^{\,k}}
+& -\ \ {_i^k \mathbb{R}_{i_p}^{k_p}}
+&\frac{ \delta_{k+k_p} [T] }{{e_{3w}}_{\,i}^{\,k+k_p}} \Bigr)^2
+& \frac{1}{4} \ {b_u}_{\,i+i_p}^{\,k} \ A_i^k \ \
+\end{matrix}
+ \right\}
+\quad \leq 0
+\end{align*}
+The last inequality is obviously obtained as we succeed in obtaining a negative summation of square quantities.
+
+Note that, if instead of multiplying $D_l^T$ by $T$, we were using another tracer field, let say $S$, then the previous demonstration would have let to:
+\begin{align*}
+\int_D S \; D_l^T \;dv &\equiv \sum_{i,k} \left\{ S \ D_l^T \ b_T \right\} \\
+&\equiv - \sum_{i,k} \sum_{\substack{i_p,\,k_p}} \left\{
+\left( \frac{ \delta_{i +i_p} [S] }{{e_{1u} }_{\,i+i_p}^{\,k}}
+ - {_i^k \mathbb{R}_{i_p}^{k_p}}
+\frac{ \delta_{k+k_p} [S] }{{e_{3w}}_{\,i}^{\,k+k_p}} \right) \right.
+\\ & \qquad \qquad \qquad \ \left.
+\left( \frac{ \delta_{i +i_p} [T] }{{e_{1u} }_{\,i+i_p}^{\,k}}
+ - {_i^k \mathbb{R}_{i_p}^{k_p}}
+\frac{ \delta_{k+k_p} [T] }{{e_{3w}}_{\,i}^{\,k+k_p}} \right)
+ \frac{1}{4} \ {b_u}_{\,i+i_p}^{\,k} \ A_i^k \
+ \right\}
+%
+\allowdisplaybreaks
+\intertext{which, by applying the same operation as before but in reverse order, leads to: }
+%
+&\equiv \sum_{i,k} \left\{ D_l^S \ T \ b_T \right\}
+\end{align*}
+This means that the iso-neutral operator is self-adjoint. There is no need to develop a specific to obtain it.
+
+
+
+$\ $\newpage %force an empty line
+% ================================================================
+% Discrete Invariants of the skew flux formulation
+% ================================================================
+\subsection{Discrete Invariants of the skew flux formulation}
+\label{Apdx_eiv_skew}
+
+
+Demonstration for the conservation of the tracer variance in the (\textbf{i},\textbf{j}) plane.
+
+This have to be moved in an Appendix.
+
+The continuous property to be demonstrated is :
+\begin{align*}
+\int_D \nabla \cdot \textbf{F}_{eiv}(T) \; T \;dv \equiv 0
+\end{align*}
+The discrete form of its left hand side is obtained using \eqref{Eq_eiv_skew}
+\begin{align*}
+ \sum\limits_{i,k} \sum_{\substack{i_p,\,k_p}} \Biggl\{ \;\;
+ \delta_i &\left[
+{e_{2u}}_{i+i_p+1/2}^{k} \;\ \ {A_{e}}_{i+i_p+1/2}^{k}
+\ \ \ { _{i+i_p+1/2}^k \mathbb{R}_{-i_p}^{k_p} } \quad \delta_{k+k_p}[T_{i+i_p+1/2}]
+ \right] \; T_i^k \\
+- \delta_k &\left[
+{e_{2u}}_i^{k+k_p+1/2} \ \ {A_{e}}_i^{k+k_p+1/2}
+\ \ { _i^{k+k_p+1/2} \mathbb{R}_{i_p}^{-k_p} } \ \ \delta_{i+i_p}[T^{k+k_p+1/2}]
+ \right] \; T_i^k \ \Biggr\}
+\end{align*}
+apply the adjoint of delta operator, it becomes
+\begin{align*}
+ \sum\limits_{i,k} \sum_{\substack{i_p,\,k_p}} \Biggl\{ \;\;
+ &\left(
+{e_{2u}}_{i+i_p+1/2}^{k} \;\ \ {A_{e}}_{i+i_p+1/2}^{k}
+\ \ \ { _{i+i_p+1/2}^k \mathbb{R}_{-i_p}^{k_p} } \quad \delta_{k+k_p}[T_{i+i_p+1/2}]
+ \right) \; \delta_{i+1/2}[T^{k}] \\
+- &\left(
+{e_{2u}}_i^{k+k_p+1/2} \ \ {A_{e}}_i^{k+k_p+1/2}
+\ \ { _i^{k+k_p+1/2} \mathbb{R}_{i_p}^{-k_p} } \ \ \delta_{i+i_p}[T^{k+k_p+1/2}]
+ \right) \; \delta_{k+1/2}[T_{i}] \ \Biggr\}
+\end{align*}
+Expending the summation on $i_p$ and $k_p$, it becomes:
+\begin{align*}
+ \begin{matrix}
+&\sum\limits_{i,k} \Bigl\{
+ &+{e_{2u}}_{i+1}^{k} &{A_{e}}_{i+1 }^{k}
+ &\ {_{i+1}^k \mathbb{R}_{- 1/2}^{-1/2}} &\delta_{k-1/2}[T_{i+1}] &\delta_{i+1/2}[T^{k}] &\\
+&&+{e_{2u}}_i^{k\ \ \ \:} &{A_{e}}_{i}^{k\ \ \ \:}
+ &\ {\ \ \;_i^k \mathbb{R}_{+1/2}^{-1/2}} &\delta_{k-1/2}[T_{i\ \ \ \;}] &\delta_{i+1/2}[T^{k}] &\\
+&&+{e_{2u}}_{i+1}^{k} &{A_{e}}_{i+1 }^{k}
+ &\ {_{i+1}^k \mathbb{R}_{- 1/2}^{+1/2}} &\delta_{k+1/2}[T_{i+1}] &\delta_{i+1/2}[T^{k}] &\\
+&&+{e_{2u}}_i^{k\ \ \ \:} &{A_{e}}_{i}^{k\ \ \ \:}
+ &\ {\ \ \;_i^k \mathbb{R}_{+1/2}^{+1/2}} &\delta_{k+1/2}[T_{i\ \ \ \;}] &\delta_{i+1/2}[T^{k}] &\\
+%
+&&-{e_{2u}}_i^{k+1} &{A_{e}}_i^{k+1}
+ &{_i^{k+1} \mathbb{R}_{-1/2}^{- 1/2}} &\delta_{i-1/2}[T^{k+1}] &\delta_{k+1/2}[T_{i}] &\\
+&&-{e_{2u}}_i^{k\ \ \ \:} &{A_{e}}_i^{k\ \ \ \:}
+ &{\ \ \;_i^k \mathbb{R}_{-1/2}^{+1/2}} &\delta_{i-1/2}[T^{k\ \ \ \:}] &\delta_{k+1/2}[T_{i}] &\\
+&&-{e_{2u}}_i^{k+1 } &{A_{e}}_i^{k+1}
+ &{_i^{k+1} \mathbb{R}_{+1/2}^{- 1/2}} &\delta_{i+1/2}[T^{k+1}] &\delta_{k+1/2}[T_{i}] &\\
+&&-{e_{2u}}_i^{k\ \ \ \:} &{A_{e}}_i^{k\ \ \ \:}
+ &{\ \ \;_i^k \mathbb{R}_{+1/2}^{+1/2}} &\delta_{i+1/2}[T^{k\ \ \ \:}] &\delta_{k+1/2}[T_{i}]
+&\Bigr\} \\
+\end{matrix}
+\end{align*}
+The two terms associated with the triad ${_i^k \mathbb{R}_{+1/2}^{+1/2}}$ are the
+same but of opposite signs, they cancel out.
+Exactly the same thing occurs for the triad ${_i^k \mathbb{R}_{-1/2}^{-1/2}}$.
+The two terms associated with the triad ${_i^k \mathbb{R}_{+1/2}^{-1/2}}$ are the
+same but both of opposite signs and shifted by 1 in $k$ direction. When summing over $k$
+they cancel out with the neighbouring grid points.
+Exactly the same thing occurs for the triad ${_i^k \mathbb{R}_{-1/2}^{+1/2}}$ in the
+$i$ direction. Therefore the sum over the domain is zero, $i.e.$ the variance of the
+tracer is preserved by the discretisation of the skew fluxes.
+
Index: /branches/UKMO/dev_r5518_GO6_under_ice_relax_dr_hook/DOC/TexFiles/Chapters/Annex_ISO.tex
===================================================================
--- /branches/UKMO/dev_r5518_GO6_under_ice_relax_dr_hook/DOC/TexFiles/Chapters/Annex_ISO.tex (revision 11643)
+++ /branches/UKMO/dev_r5518_GO6_under_ice_relax_dr_hook/DOC/TexFiles/Chapters/Annex_ISO.tex (revision 11643)
@@ -0,0 +1,1194 @@
+% ================================================================
+% Iso-neutral diffusion :
+% ================================================================
+\chapter[Iso-neutral diffusion and eddy advection using
+triads]{Iso-neutral diffusion and eddy advection using triads}
+\label{sec:triad}
+\minitoc
+\pagebreak
+\section{Choice of \ngn{namtra\_ldf} namelist parameters}
+%-----------------------------------------nam_traldf------------------------------------------------------
+\namdisplay{namtra_ldf}
+%---------------------------------------------------------------------------------------------------------
+If the namelist variable \np{ln\_traldf\_grif} is set true (and
+\key{ldfslp} is set), \NEMO updates both active and passive tracers
+using the Griffies triad representation of iso-neutral diffusion and
+the eddy-induced advective skew (GM) fluxes. Otherwise (by default) the
+filtered version of Cox's original scheme is employed
+(\S\ref{LDF_slp}). In the present implementation of the Griffies
+scheme, the advective skew fluxes are implemented even if
+\key{traldf\_eiv} is not set.
+
+Values of iso-neutral diffusivity and GM coefficient are set as
+described in \S\ref{LDF_coef}. If none of the keys \key{traldf\_cNd},
+N=1,2,3 is set (the default), spatially constant iso-neutral $A_l$ and
+GM diffusivity $A_e$ are directly set by \np{rn\_aeih\_0} and
+\np{rn\_aeiv\_0}. If 2D-varying coefficients are set with
+\key{traldf\_c2d} then $A_l$ is reduced in proportion with horizontal
+scale factor according to \eqref{Eq_title} \footnote{Except in global ORCA
+ $0.5^{\circ}$ runs with \key{traldf\_eiv}, where
+ $A_l$ is set like $A_e$ but with a minimum vale of
+ $100\;\mathrm{m}^2\;\mathrm{s}^{-1}$}. In idealised setups with
+\key{traldf\_c2d}, $A_e$ is reduced similarly, but if \key{traldf\_eiv}
+is set in the global configurations with \key{traldf\_c2d}, a horizontally varying $A_e$ is
+instead set from the Held-Larichev parameterisation\footnote{In this
+ case, $A_e$ at low latitudes $|\theta|<20^{\circ}$ is further
+ reduced by a factor $|f/f_{20}|$, where $f_{20}$ is the value of $f$
+ at $20^{\circ}$~N} (\mdl{ldfeiv}) and \np{rn\_aeiv\_0} is ignored
+unless it is zero.
+
+The options specific to the Griffies scheme include:
+\begin{description}[font=\normalfont]
+\item[\np{ln\_traldf\_gdia}] Default value is false. See \S\ref{sec:triad:sfdiag}. If this is set true, time-mean
+ eddy-advective (GM) velocities are output for diagnostic purposes, even
+ though the eddy advection is accomplished by means of the skew
+ fluxes.
+\item[\np{ln\_traldf\_iso}] See \S\ref{sec:triad:taper}. If this is set false (the default), then
+ `iso-neutral' mixing is accomplished within the surface mixed-layer
+ along slopes linearly decreasing with depth from the value immediately below
+ the mixed-layer to zero (flat) at the surface (\S\ref{sec:triad:lintaper}). This is the same
+ treatment as used in the default implementation
+ \S\ref{LDF_slp_iso}; Fig.~\ref{Fig_eiv_slp}. Where
+ \np{ln\_traldf\_iso} is set true, the vertical skew flux is further
+ reduced to ensure no vertical buoyancy flux, giving an almost pure
+ horizontal diffusive tracer flux within the mixed layer. This is similar to
+ the tapering suggested by \citet{Gerdes1991}. See \S\ref{sec:triad:Gerdes-taper}
+\item[\np{ln\_traldf\_botmix}] See \S\ref{sec:triad:iso_bdry}. If this
+ is set false (the default) then the lateral diffusive fluxes
+ associated with triads partly masked by topography are neglected. If
+ it is set true, however, then these lateral diffusive fluxes are
+ applied, giving smoother bottom tracer fields at the cost of
+ introducing diapycnal mixing.
+\end{description}
+\section{Triad formulation of iso-neutral diffusion}
+\label{sec:triad:iso}
+We have implemented into \NEMO a scheme inspired by \citet{Griffies_al_JPO98}, but formulated within the \NEMO
+framework, using scale factors rather than grid-sizes.
+
+\subsection{The iso-neutral diffusion operator}
+The iso-neutral second order tracer diffusive operator for small
+angles between iso-neutral surfaces and geopotentials is given by
+\eqref{Eq_PE_iso_tensor}:
+\begin{subequations} \label{eq:triad:PE_iso_tensor}
+ \begin{equation}
+ D^{lT}=-\Div\vect{f}^{lT}\equiv
+ -\frac{1}{e_1e_2e_3}\left[\pd{i}\left (f_1^{lT}e_2e_3\right) +
+ \pd{j}\left (f_2^{lT}e_2e_3\right) + \pd{k}\left (f_3^{lT}e_1e_2\right)\right],
+ \end{equation}
+ where the diffusive flux per unit area of physical space
+ \begin{equation}
+ \vect{f}^{lT}=-\Alt\Re\cdot\grad T,
+ \end{equation}
+ \begin{equation}
+ \label{eq:triad:PE_iso_tensor:c}
+ \mbox{with}\quad \;\;\Re =
+ \begin{pmatrix}
+ 1&0&-r_1\mystrut \\
+ 0&1&-r_2\mystrut \\
+ -r_1&-r_2&r_1 ^2+r_2 ^2\mystrut
+ \end{pmatrix}
+ \quad \text{and} \quad\grad T=
+ \begin{pmatrix}
+ \frac{1}{e_1}\pd[T]{i}\mystrut \\
+ \frac{1}{e_2}\pd[T]{j}\mystrut \\
+ \frac{1}{e_3}\pd[T]{k}\mystrut
+ \end{pmatrix}.
+ \end{equation}
+\end{subequations}
+% \left( {{\begin{array}{*{20}c}
+% 1 \hfill & 0 \hfill & {-r_1 } \hfill \\
+% 0 \hfill & 1 \hfill & {-r_2 } \hfill \\
+% {-r_1 } \hfill & {-r_2 } \hfill & {r_1 ^2+r_2 ^2} \hfill \\
+% \end{array} }} \right)
+ Here \eqref{Eq_PE_iso_slopes}
+\begin{align*}
+ r_1 &=-\frac{e_3 }{e_1 } \left( \frac{\partial \rho }{\partial i}
+ \right)
+ \left( {\frac{\partial \rho }{\partial k}} \right)^{-1} \\
+ &=-\frac{e_3 }{e_1 } \left( -\alpha\frac{\partial T }{\partial i} +
+ \beta\frac{\partial S }{\partial i} \right) \left(
+ -\alpha\frac{\partial T }{\partial k} + \beta\frac{\partial S
+ }{\partial k} \right)^{-1}
+\end{align*}
+is the $i$-component of the slope of the iso-neutral surface relative to the computational
+surface, and $r_2$ is the $j$-component.
+
+We will find it useful to consider the fluxes per unit area in $i,j,k$
+space; we write
+\begin{equation}
+ \label{eq:triad:Fijk}
+ \vect{F}_{\mathrm{iso}}=\left(f_1^{lT}e_2e_3, f_2^{lT}e_1e_3, f_3^{lT}e_1e_2\right).
+\end{equation}
+Additionally, we will sometimes write the contributions towards the
+fluxes $\vect{f}$ and $\vect{F}_\mathrm{iso}$ from the component
+$R_{ij}$ of $\Re$ as $f_{ij}$, $F_{\mathrm{iso}\: ij}$, with
+$f_{ij}=R_{ij}e_i^{-1}\partial T/\partial x_i$ (no summation) etc.
+
+The off-diagonal terms of the small angle diffusion tensor
+\eqref{Eq_PE_iso_tensor}, \eqref{eq:triad:PE_iso_tensor:c} produce skew-fluxes along the
+$i$- and $j$-directions resulting from the vertical tracer gradient:
+\begin{align}
+ \label{eq:triad:i13c}
+ f_{13}=&+\Alt r_1\frac{1}{e_3}\frac{\partial T}{\partial k},\qquad f_{23}=+\Alt r_2\frac{1}{e_3}\frac{\partial T}{\partial k}\\
+\intertext{and in the k-direction resulting from the lateral tracer gradients}
+ \label{eq:triad:i31c}
+ f_{31}+f_{32}=& \Alt r_1\frac{1}{e_1}\frac{\partial T}{\partial i}+\Alt r_2\frac{1}{e_1}\frac{\partial T}{\partial i}
+\end{align}
+
+The vertical diffusive flux associated with the $_{33}$
+component of the small angle diffusion tensor is
+\begin{equation}
+ \label{eq:triad:i33c}
+ f_{33}=-\Alt(r_1^2 +r_2^2) \frac{1}{e_3}\frac{\partial T}{\partial k}.
+\end{equation}
+
+Since there are no cross terms involving $r_1$ and $r_2$ in the above, we can
+consider the iso-neutral diffusive fluxes separately in the $i$-$k$ and $j$-$k$
+planes, just adding together the vertical components from each
+plane. The following description will describe the fluxes on the $i$-$k$
+plane.
+
+There is no natural discretization for the $i$-component of the
+skew-flux, \eqref{eq:triad:i13c}, as
+although it must be evaluated at $u$-points, it involves vertical
+gradients (both for the tracer and the slope $r_1$), defined at
+$w$-points. Similarly, the vertical skew flux, \eqref{eq:triad:i31c}, is evaluated at
+$w$-points but involves horizontal gradients defined at $u$-points.
+
+\subsection{The standard discretization}
+The straightforward approach to discretize the lateral skew flux
+\eqref{eq:triad:i13c} from tracer cell $i,k$ to $i+1,k$, introduced in 1995
+into OPA, \eqref{Eq_tra_ldf_iso}, is to calculate a mean vertical
+gradient at the $u$-point from the average of the four surrounding
+vertical tracer gradients, and multiply this by a mean slope at the
+$u$-point, calculated from the averaged surrounding vertical density
+gradients. The total area-integrated skew-flux (flux per unit area in
+$ijk$ space) from tracer cell $i,k$
+to $i+1,k$, noting that the $e_{{3}_{i+1/2}^k}$ in the area
+$e{_{3}}_{i+1/2}^k{e_{2}}_{i+1/2}i^k$ at the $u$-point cancels out with
+the $1/{e_{3}}_{i+1/2}^k$ associated with the vertical tracer
+gradient, is then \eqref{Eq_tra_ldf_iso}
+\begin{equation*}
+ \left(F_u^{13} \right)_{i+\hhalf}^k = \Alts_{i+\hhalf}^k
+ {e_{2}}_{i+1/2}^k \overline{\overline
+ r_1} ^{\,i,k}\,\overline{\overline{\delta_k T}}^{\,i,k},
+\end{equation*}
+where
+\begin{equation*}
+ \overline{\overline
+ r_1} ^{\,i,k} = -\frac{{e_{3u}}_{i+1/2}^k}{{e_{1u}}_{i+1/2}^k}
+ \frac{\delta_{i+1/2} [\rho]}{\overline{\overline{\delta_k \rho}}^{\,i,k}},
+\end{equation*}
+and here and in the following we drop the $^{lT}$ superscript from
+$\Alt$ for simplicity.
+Unfortunately the resulting combination $\overline{\overline{\delta_k
+ \bullet}}^{\,i,k}$ of a $k$ average and a $k$ difference %of the tracer
+reduces to $\bullet_{k+1}-\bullet_{k-1}$, so two-grid-point oscillations are
+invisible to this discretization of the iso-neutral operator. These
+\emph{computational modes} will not be damped by this operator, and
+may even possibly be amplified by it. Consequently, applying this
+operator to a tracer does not guarantee the decrease of its
+global-average variance. To correct this, we introduced a smoothing of
+the slopes of the iso-neutral surfaces (see \S\ref{LDF}). This
+technique works for $T$ and $S$ in so far as they are active tracers
+($i.e.$ they enter the computation of density), but it does not work
+for a passive tracer.
+\subsection{Expression of the skew-flux in terms of triad slopes}
+\citep{Griffies_al_JPO98} introduce a different discretization of the
+off-diagonal terms that nicely solves the problem.
+% Instead of multiplying the mean slope calculated at the $u$-point by
+% the mean vertical gradient at the $u$-point,
+% >>>>>>>>>>>>>>>>>>>>>>>>>>>>
+\begin{figure}[h] \begin{center}
+ \includegraphics[width=1.05\textwidth]{./TexFiles/Figures/Fig_GRIFF_triad_fluxes}
+ \caption{ \label{fig:triad:ISO_triad}
+ (a) Arrangement of triads $S_i$ and tracer gradients to
+ give lateral tracer flux from box $i,k$ to $i+1,k$
+ (b) Triads $S'_i$ and tracer gradients to give vertical tracer flux from
+ box $i,k$ to $i,k+1$.}
+ \end{center} \end{figure}
+% >>>>>>>>>>>>>>>>>>>>>>>>>>>>
+They get the skew flux from the products of the vertical gradients at
+each $w$-point surrounding the $u$-point with the corresponding `triad'
+slope calculated from the lateral density gradient across the $u$-point
+divided by the vertical density gradient at the same $w$-point as the
+tracer gradient. See Fig.~\ref{fig:triad:ISO_triad}a, where the thick lines
+denote the tracer gradients, and the thin lines the corresponding
+triads, with slopes $s_1, \dotsc s_4$. The total area-integrated
+skew-flux from tracer cell $i,k$ to $i+1,k$
+\begin{multline}
+ \label{eq:triad:i13}
+ \left( F_u^{13} \right)_{i+\frac{1}{2}}^k = \Alts_{i+1}^k a_1 s_1
+ \delta _{k+\frac{1}{2}} \left[ T^{i+1}
+ \right]/e_{{3w}_{i+1}}^{k+\frac{1}{2}} + \Alts _i^k a_2 s_2 \delta
+ _{k+\frac{1}{2}} \left[ T^i
+ \right]/e_{{3w}_{i+1}}^{k+\frac{1}{2}} \\
+ +\Alts _{i+1}^k a_3 s_3 \delta _{k-\frac{1}{2}} \left[ T^{i+1}
+ \right]/e_{{3w}_{i+1}}^{k+\frac{1}{2}} +\Alts _i^k a_4 s_4 \delta
+ _{k-\frac{1}{2}} \left[ T^i \right]/e_{{3w}_{i+1}}^{k+\frac{1}{2}},
+\end{multline}
+where the contributions of the triad fluxes are weighted by areas
+$a_1, \dotsc a_4$, and $\Alts$ is now defined at the tracer points
+rather than the $u$-points. This discretization gives a much closer
+stencil, and disallows the two-point computational modes.
+
+ The vertical skew flux \eqref{eq:triad:i31c} from tracer cell $i,k$ to $i,k+1$ at the
+$w$-point $i,k+\hhalf$ is constructed similarly (Fig.~\ref{fig:triad:ISO_triad}b)
+by multiplying lateral tracer gradients from each of the four
+surrounding $u$-points by the appropriate triad slope:
+\begin{multline}
+ \label{eq:triad:i31}
+ \left( F_w^{31} \right) _i ^{k+\frac{1}{2}} = \Alts_i^{k+1} a_{1}'
+ s_{1}' \delta _{i-\frac{1}{2}} \left[ T^{k+1} \right]/{e_{3u}}_{i-\frac{1}{2}}^{k+1}
+ +\Alts_i^{k+1} a_{2}' s_{2}' \delta _{i+\frac{1}{2}} \left[ T^{k+1} \right]/{e_{3u}}_{i+\frac{1}{2}}^{k+1}\\
+ + \Alts_i^k a_{3}' s_{3}' \delta _{i-\frac{1}{2}} \left[ T^k\right]/{e_{3u}}_{i-\frac{1}{2}}^k
+ +\Alts_i^k a_{4}' s_{4}' \delta _{i+\frac{1}{2}} \left[ T^k \right]/{e_{3u}}_{i+\frac{1}{2}}^k.
+\end{multline}
+
+We notate the triad slopes $s_i$ and $s'_i$ in terms of the `anchor point' $i,k$
+(appearing in both the vertical and lateral gradient), and the $u$- and
+$w$-points $(i+i_p,k)$, $(i,k+k_p)$ at the centres of the `arms' of the
+triad as follows (see also Fig.~\ref{fig:triad:ISO_triad}):
+\begin{equation}
+ \label{eq:triad:R}
+ _i^k \mathbb{R}_{i_p}^{k_p}
+ =-\frac{ {e_{3w}}_{\,i}^{\,k+k_p}} { {e_{1u}}_{\,i+i_p}^{\,k}}
+ \
+ \frac
+ {\left(\alpha / \beta \right)_i^k \ \delta_{i + i_p}[T^k] - \delta_{i + i_p}[S^k] }
+ {\left(\alpha / \beta \right)_i^k \ \delta_{k+k_p}[T^i ] - \delta_{k+k_p}[S^i ] }.
+\end{equation}
+In calculating the slopes of the local neutral
+surfaces, the expansion coefficients $\alpha$ and $\beta$ are
+evaluated at the anchor points of the triad \footnote{Note that in \eqref{eq:triad:R} we use the ratio $\alpha / \beta$
+instead of multiplying the temperature derivative by $\alpha$ and the
+salinity derivative by $\beta$. This is more efficient as the ratio
+$\alpha / \beta$ can to be evaluated directly}, while the metrics are
+calculated at the $u$- and $w$-points on the arms.
+
+% >>>>>>>>>>>>>>>>>>>>>>>>>>>>
+\begin{figure}[h] \begin{center}
+ \includegraphics[width=0.80\textwidth]{./TexFiles/Figures/Fig_GRIFF_qcells}
+ \caption{ \label{fig:triad:qcells}
+ Triad notation for quarter cells. $T$-cells are inside
+ boxes, while the $i+\half,k$ $u$-cell is shaded in green and the
+ $i,k+\half$ $w$-cell is shaded in pink.}
+ \end{center} \end{figure}
+% >>>>>>>>>>>>>>>>>>>>>>>>>>>>
+
+Each triad $\{_i^k\:_{i_p}^{k_p}\}$ is associated (Fig.~\ref{fig:triad:qcells}) with the quarter
+cell that is the intersection of the $i,k$ $T$-cell, the $i+i_p,k$
+$u$-cell and the $i,k+k_p$ $w$-cell. Expressing the slopes $s_i$ and
+$s'_i$ in \eqref{eq:triad:i13} and \eqref{eq:triad:i31} in this notation, we have
+e.g.\ $s_1=s'_1={\:}_i^k \mathbb{R}_{1/2}^{1/2}$. Each triad slope $_i^k
+\mathbb{R}_{i_p}^{k_p}$ is used once (as an $s$) to calculate the
+lateral flux along its $u$-arm, at $(i+i_p,k)$, and then again as an
+$s'$ to calculate the vertical flux along its $w$-arm at
+$(i,k+k_p)$. Each vertical area $a_i$ used to calculate the lateral
+flux and horizontal area $a'_i$ used to calculate the vertical flux
+can also be identified as the area across the $u$- and $w$-arms of a
+unique triad, and we notate these areas, similarly to the triad
+slopes, as $_i^k{\mathbb{A}_u}_{i_p}^{k_p}$,
+$_i^k{\mathbb{A}_w}_{i_p}^{k_p}$, where e.g. in \eqref{eq:triad:i13}
+$a_{1}={\:}_i^k{\mathbb{A}_u}_{1/2}^{1/2}$, and in \eqref{eq:triad:i31}
+$a'_{1}={\:}_i^k{\mathbb{A}_w}_{1/2}^{1/2}$.
+
+\subsection{The full triad fluxes}
+A key property of iso-neutral diffusion is that it should not affect
+the (locally referenced) density. In particular there should be no
+lateral or vertical density flux. The lateral density flux disappears so long as the
+area-integrated lateral diffusive flux from tracer cell $i,k$ to
+$i+1,k$ coming from the $_{11}$ term of the diffusion tensor takes the
+form
+\begin{equation}
+ \label{eq:triad:i11}
+ \left( F_u^{11} \right) _{i+\frac{1}{2}} ^{k} =
+ - \left( \Alts_i^{k+1} a_{1} + \Alts_i^{k+1} a_{2} + \Alts_i^k
+ a_{3} + \Alts_i^k a_{4} \right)
+ \frac{\delta _{i+1/2} \left[ T^k\right]}{{e_{1u}}_{\,i+1/2}^{\,k}},
+\end{equation}
+where the areas $a_i$ are as in \eqref{eq:triad:i13}. In this case,
+separating the total lateral flux, the sum of \eqref{eq:triad:i13} and
+\eqref{eq:triad:i11}, into triad components, a lateral tracer
+flux
+\begin{equation}
+ \label{eq:triad:latflux-triad}
+ _i^k {\mathbb{F}_u}_{i_p}^{k_p} (T) = - \Alts_i^k{ \:}_i^k{\mathbb{A}_u}_{i_p}^{k_p}
+ \left(
+ \frac{ \delta_{i+ i_p}[T^k] }{ {e_{1u}}_{\,i + i_p}^{\,k} }
+ -\ {_i^k\mathbb{R}_{i_p}^{k_p}} \
+ \frac{ \delta_{k+k_p} [T^i] }{{e_{3w}}_{\,i}^{\,k+k_p} }
+ \right)
+\end{equation}
+can be identified with each triad. Then, because the
+same metric factors ${e_{3w}}_{\,i}^{\,k+k_p}$ and
+${e_{1u}}_{\,i+i_p}^{\,k}$ are employed for both the density gradients
+in $ _i^k \mathbb{R}_{i_p}^{k_p}$ and the tracer gradients, the lateral
+density flux associated with each triad separately disappears.
+\begin{equation}
+ \label{eq:triad:latflux-rho}
+ {\mathbb{F}_u}_{i_p}^{k_p} (\rho)=-\alpha _i^k {\:}_i^k {\mathbb{F}_u}_{i_p}^{k_p} (T) + \beta_i^k {\:}_i^k {\mathbb{F}_u}_{i_p}^{k_p} (S)=0
+\end{equation}
+Thus the total flux $\left( F_u^{31} \right) ^i _{i,k+\frac{1}{2}} +
+\left( F_u^{11} \right) ^i _{i,k+\frac{1}{2}}$ from tracer cell $i,k$
+to $i+1,k$ must also vanish since it is a sum of four such triad fluxes.
+
+The squared slope $r_1^2$ in the expression \eqref{eq:triad:i33c} for the
+$_{33}$ component is also expressed in terms of area-weighted
+squared triad slopes, so the area-integrated vertical flux from tracer
+cell $i,k$ to $i,k+1$ resulting from the $r_1^2$ term is
+\begin{equation}
+ \label{eq:triad:i33}
+ \left( F_w^{33} \right) _i^{k+\frac{1}{2}} =
+ - \left( \Alts_i^{k+1} a_{1}' s_{1}'^2
+ + \Alts_i^{k+1} a_{2}' s_{2}'^2
+ + \Alts_i^k a_{3}' s_{3}'^2
+ + \Alts_i^k a_{4}' s_{4}'^2 \right)\delta_{k+\frac{1}{2}} \left[ T^{i+1} \right],
+\end{equation}
+where the areas $a'$ and slopes $s'$ are the same as in
+\eqref{eq:triad:i31}.
+Then, separating the total vertical flux, the sum of \eqref{eq:triad:i31} and
+\eqref{eq:triad:i33}, into triad components, a vertical flux
+\begin{align}
+ \label{eq:triad:vertflux-triad}
+ _i^k {\mathbb{F}_w}_{i_p}^{k_p} (T)
+ &= \Alts_i^k{\: }_i^k{\mathbb{A}_w}_{i_p}^{k_p}
+ \left(
+ {_i^k\mathbb{R}_{i_p}^{k_p}}\frac{ \delta_{i+ i_p}[T^k] }{ {e_{1u}}_{\,i + i_p}^{\,k} }
+ -\ \left({_i^k\mathbb{R}_{i_p}^{k_p}}\right)^2 \
+ \frac{ \delta_{k+k_p} [T^i] }{{e_{3w}}_{\,i}^{\,k+k_p} }
+ \right) \\
+ &= - \left(\left.{ }_i^k{\mathbb{A}_w}_{i_p}^{k_p}\right/{ }_i^k{\mathbb{A}_u}_{i_p}^{k_p}\right)
+ {_i^k\mathbb{R}_{i_p}^{k_p}}{\: }_i^k{\mathbb{F}_u}_{i_p}^{k_p} (T) \label{eq:triad:vertflux-triad2}
+\end{align}
+may be associated with each triad. Each vertical density flux $_i^k {\mathbb{F}_w}_{i_p}^{k_p} (\rho)$
+associated with a triad then separately disappears (because the
+lateral flux $_i^k{\mathbb{F}_u}_{i_p}^{k_p} (\rho)$
+disappears). Consequently the total vertical density flux $\left( F_w^{31} \right)_i ^{k+\frac{1}{2}} +
+\left( F_w^{33} \right)_i^{k+\frac{1}{2}}$ from tracer cell $i,k$
+to $i,k+1$ must also vanish since it is a sum of four such triad
+fluxes.
+
+We can explicitly identify (Fig.~\ref{fig:triad:qcells}) the triads associated with the $s_i$, $a_i$, and $s'_i$, $a'_i$ used in the definition of
+the $u$-fluxes and $w$-fluxes in
+\eqref{eq:triad:i31}, \eqref{eq:triad:i13}, \eqref{eq:triad:i11} \eqref{eq:triad:i33} and
+Fig.~\ref{fig:triad:ISO_triad} to write out the iso-neutral fluxes at $u$- and
+$w$-points as sums of the triad fluxes that cross the $u$- and $w$-faces:
+%(Fig.~\ref{Fig_ISO_triad}):
+\begin{flalign} \label{Eq_iso_flux} \vect{F}_\mathrm{iso}(T) &\equiv
+ \sum_{\substack{i_p,\,k_p}}
+ \begin{pmatrix}
+ {_{i+1/2-i_p}^k {\mathbb{F}_u}_{i_p}^{k_p} } (T) \\
+ \\
+ {_i^{k+1/2-k_p} {\mathbb{F}_w}_{i_p}^{k_p} } (T) \\
+ \end{pmatrix}.
+\end{flalign}
+\subsection{Ensuring the scheme does not increase tracer variance}
+\label{sec:triad:variance}
+
+We now require that this operator should not increase the
+globally-integrated tracer variance.
+%This changes according to
+% \begin{align*}
+% &\int_D D_l^T \; T \;dv \equiv \sum_{i,k} \left\{ T \ D_l^T \ b_T \right\} \\
+% &\equiv + \sum_{i,k} \sum_{\substack{i_p,\,k_p}} \left\{
+% \delta_{i} \left[{_{i+1/2-i_p}^k {\mathbb{F}_u }_{i_p}^{k_p}} \right]
+% + \delta_{k} \left[ {_i^{k+1/2-k_p} {\mathbb{F}_w}_{i_p}^{k_p}} \right] \ T \right\} \\
+% &\equiv - \sum_{i,k} \sum_{\substack{i_p,\,k_p}} \left\{
+% {_{i+1/2-i_p}^k {\mathbb{F}_u }_{i_p}^{k_p}} \ \delta_{i+1/2} [T]
+% + {_i^{k+1/2-k_p} {\mathbb{F}_w}_{i_p}^{k_p}} \ \delta_{k+1/2} [T] \right\} \\
+% \end{align*}
+Each triad slope $_i^k\mathbb{R}_{i_p}^{k_p}$ drives a lateral flux
+$_i^k{\mathbb{F}_u}_{i_p}^{k_p} (T)$ across the $u$-point $i+i_p,k$ and
+a vertical flux $_i^k{\mathbb{F}_w}_{i_p}^{k_p} (T)$ across the
+$w$-point $i,k+k_p$. The lateral flux drives a net rate of change of
+variance, summed over the two $T$-points $i+i_p-\half,k$ and $i+i_p+\half,k$, of
+\begin{multline}
+ {b_T}_{i+i_p-1/2}^k\left(\frac{\partial T}{\partial t}T\right)_{i+i_p-1/2}^k+
+ \quad {b_T}_{i+i_p+1/2}^k\left(\frac{\partial T}{\partial
+ t}T\right)_{i+i_p+1/2}^k \\
+ \begin{split}
+ &= -T_{i+i_p-1/2}^k{\;} _i^k{\mathbb{F}_u}_{i_p}^{k_p} (T) \quad + \quad T_{i+i_p+1/2}^k
+ {\;}_i^k{\mathbb{F}_u}_{i_p}^{k_p} (T) \\
+ &={\;} _i^k{\mathbb{F}_u}_{i_p}^{k_p} (T)\,\delta_{i+ i_p}[T^k], \label{eq:triad:dvar_iso_i}
+ \end{split}
+\end{multline}
+while the vertical flux similarly drives a net rate of change of
+variance summed over the $T$-points $i,k+k_p-\half$ (above) and
+$i,k+k_p+\half$ (below) of
+\begin{equation}
+\label{eq:triad:dvar_iso_k}
+ _i^k{\mathbb{F}_w}_{i_p}^{k_p} (T) \,\delta_{k+ k_p}[T^i].
+\end{equation}
+The total variance tendency driven by the triad is the sum of these
+two. Expanding $_i^k{\mathbb{F}_u}_{i_p}^{k_p} (T)$ and
+$_i^k{\mathbb{F}_w}_{i_p}^{k_p} (T)$ with \eqref{eq:triad:latflux-triad} and
+\eqref{eq:triad:vertflux-triad}, it is
+\begin{multline*}
+-\Alts_i^k\left \{
+{ } _i^k{\mathbb{A}_u}_{i_p}^{k_p}
+ \left(
+ \frac{ \delta_{i+ i_p}[T^k] }{ {e_{1u}}_{\,i + i_p}^{\,k} }
+ - {_i^k\mathbb{R}_{i_p}^{k_p}} \
+ \frac{ \delta_{k+k_p} [T^i] }{{e_{3w}}_{\,i}^{\,k+k_p} }\right)\,\delta_{i+ i_p}[T^k] \right.\\
+- \left. { } _i^k{\mathbb{A}_w}_{i_p}^{k_p}
+ \left(
+ \frac{ \delta_{i+ i_p}[T^k] }{ {e_{1u}}_{\,i + i_p}^{\,k} }
+ -{\:}_i^k\mathbb{R}_{i_p}^{k_p}
+ \frac{ \delta_{k+k_p} [T^i] }{{e_{3w}}_{\,i}^{\,k+k_p} }
+ \right) {\,}_i^k\mathbb{R}_{i_p}^{k_p}\delta_{k+ k_p}[T^i]
+\right \}.
+\end{multline*}
+The key point is then that if we require
+$_i^k{\mathbb{A}_u}_{i_p}^{k_p}$ and $_i^k{\mathbb{A}_w}_{i_p}^{k_p}$
+to be related to a triad volume $_i^k\mathbb{V}_{i_p}^{k_p}$ by
+\begin{equation}
+ \label{eq:triad:V-A}
+ _i^k\mathbb{V}_{i_p}^{k_p}
+ ={\;}_i^k{\mathbb{A}_u}_{i_p}^{k_p}\,{e_{1u}}_{\,i + i_p}^{\,k}
+ ={\;}_i^k{\mathbb{A}_w}_{i_p}^{k_p}\,{e_{3w}}_{\,i}^{\,k + k_p},
+\end{equation}
+the variance tendency reduces to the perfect square
+\begin{equation}
+ \label{eq:triad:perfect-square}
+ -\Alts_i^k{\:} _i^k\mathbb{V}_{i_p}^{k_p}
+ \left(
+ \frac{ \delta_{i+ i_p}[T^k] }{ {e_{1u}}_{\,i + i_p}^{\,k} }
+ -{\:}_i^k\mathbb{R}_{i_p}^{k_p}
+ \frac{ \delta_{k+k_p} [T^i] }{{e_{3w}}_{\,i}^{\,k+k_p} }
+ \right)^2\leq 0.
+\end{equation}
+Thus, the constraint \eqref{eq:triad:V-A} ensures that the fluxes (\ref{eq:triad:latflux-triad}, \ref{eq:triad:vertflux-triad}) associated
+with a given slope triad $_i^k\mathbb{R}_{i_p}^{k_p}$ do not increase
+the net variance. Since the total fluxes are sums of such fluxes from
+the various triads, this constraint, applied to all triads, is
+sufficient to ensure that the globally integrated variance does not
+increase.
+
+The expression \eqref{eq:triad:V-A} can be interpreted as a discretization
+of the global integral
+\begin{equation}
+ \label{eq:triad:cts-var}
+ \frac{\partial}{\partial t}\int\!\half T^2\, dV =
+ \int\!\mathbf{F}\cdot\nabla T\, dV,
+\end{equation}
+where, within each triad volume $_i^k\mathbb{V}_{i_p}^{k_p}$, the
+lateral and vertical fluxes/unit area
+\[
+\mathbf{F}=\left(
+\left.{}_i^k{\mathbb{F}_u}_{i_p}^{k_p} (T)\right/{}_i^k{\mathbb{A}_u}_{i_p}^{k_p},
+\left.{\:}_i^k{\mathbb{F}_w}_{i_p}^{k_p} (T)\right/{}_i^k{\mathbb{A}_w}_{i_p}^{k_p}
+ \right)
+\]
+and the gradient
+ \[\nabla T = \left(
+\left.\delta_{i+ i_p}[T^k] \right/ {e_{1u}}_{\,i + i_p}^{\,k},
+\left.\delta_{k+ k_p}[T^i] \right/ {e_{3w}}_{\,i}^{\,k + k_p}
+\right)
+\]
+\subsection{Triad volumes in Griffes's scheme and in \NEMO}
+To complete the discretization we now need only specify the triad
+volumes $_i^k\mathbb{V}_{i_p}^{k_p}$. \citet{Griffies_al_JPO98} identify
+these $_i^k\mathbb{V}_{i_p}^{k_p}$ as the volumes of the quarter
+cells, defined in terms of the distances between $T$, $u$,$f$ and
+$w$-points. This is the natural discretization of
+\eqref{eq:triad:cts-var}. The \NEMO model, however, operates with scale
+factors instead of grid sizes, and scale factors for the quarter
+cells are not defined. Instead, therefore we simply choose
+\begin{equation}
+ \label{eq:triad:V-NEMO}
+ _i^k\mathbb{V}_{i_p}^{k_p}=\quarter {b_u}_{i+i_p}^k,
+\end{equation}
+as a quarter of the volume of the $u$-cell inside which the triad
+quarter-cell lies. This has the nice property that when the slopes
+$\mathbb{R}$ vanish, the lateral flux from tracer cell $i,k$ to
+$i+1,k$ reduces to the classical form
+\begin{equation}
+ \label{eq:triad:lat-normal}
+-\overline\Alts_{\,i+1/2}^k\;
+\frac{{b_u}_{i+1/2}^k}{{e_{1u}}_{\,i + i_p}^{\,k}}
+\;\frac{\delta_{i+ 1/2}[T^k] }{{e_{1u}}_{\,i + i_p}^{\,k}}
+ = -\overline\Alts_{\,i+1/2}^k\;\frac{{e_{1w}}_{\,i + 1/2}^{\,k}\:{e_{1v}}_{\,i + 1/2}^{\,k}\;\delta_{i+ 1/2}[T^k]}{{e_{1u}}_{\,i + 1/2}^{\,k}}.
+\end{equation}
+In fact if the diffusive coefficient is defined at $u$-points, so that
+we employ $\Alts_{i+i_p}^k$ instead of $\Alts_i^k$ in the definitions of the
+triad fluxes \eqref{eq:triad:latflux-triad} and \eqref{eq:triad:vertflux-triad},
+we can replace $\overline{A}_{\,i+1/2}^k$ by $A_{i+1/2}^k$ in the above.
+
+\subsection{Summary of the scheme}
+The iso-neutral fluxes at $u$- and
+$w$-points are the sums of the triad fluxes that cross the $u$- and
+$w$-faces \eqref{Eq_iso_flux}:
+\begin{subequations}\label{eq:triad:alltriadflux}
+ \begin{flalign}\label{eq:triad:vect_isoflux}
+ \vect{F}_\mathrm{iso}(T) &\equiv
+ \sum_{\substack{i_p,\,k_p}}
+ \begin{pmatrix}
+ {_{i+1/2-i_p}^k {\mathbb{F}_u}_{i_p}^{k_p} } (T) \\
+ \\
+ {_i^{k+1/2-k_p} {\mathbb{F}_w}_{i_p}^{k_p} } (T)
+ \end{pmatrix},
+ \end{flalign}
+ where \eqref{eq:triad:latflux-triad}:
+ \begin{align}
+ \label{eq:triad:triadfluxu}
+ _i^k {\mathbb{F}_u}_{i_p}^{k_p} (T) &= - \Alts_i^k{
+ \:}\frac{{{}_i^k\mathbb{V}}_{i_p}^{k_p}}{{e_{1u}}_{\,i + i_p}^{\,k}}
+ \left(
+ \frac{ \delta_{i+ i_p}[T^k] }{ {e_{1u}}_{\,i + i_p}^{\,k} }
+ -\ {_i^k\mathbb{R}_{i_p}^{k_p}} \
+ \frac{ \delta_{k+k_p} [T^i] }{{e_{3w}}_{\,i}^{\,k+k_p} }
+ \right),\\
+ \intertext{and}
+ _i^k {\mathbb{F}_w}_{i_p}^{k_p} (T)
+ &= \Alts_i^k{\: }\frac{{{}_i^k\mathbb{V}}_{i_p}^{k_p}}{{e_{3w}}_{\,i}^{\,k+k_p}}
+ \left(
+ {_i^k\mathbb{R}_{i_p}^{k_p}}\frac{ \delta_{i+ i_p}[T^k] }{ {e_{1u}}_{\,i + i_p}^{\,k} }
+ -\ \left({_i^k\mathbb{R}_{i_p}^{k_p}}\right)^2 \
+ \frac{ \delta_{k+k_p} [T^i] }{{e_{3w}}_{\,i}^{\,k+k_p} }
+ \right),\label{eq:triad:triadfluxw}
+ \end{align}
+ with \eqref{eq:triad:V-NEMO}
+ \begin{equation}
+ \label{eq:triad:V-NEMO2}
+ _i^k{\mathbb{V}}_{i_p}^{k_p}=\quarter {b_u}_{i+i_p}^k.
+ \end{equation}
+\end{subequations}
+
+ The divergence of the expression \eqref{Eq_iso_flux} for the fluxes gives the iso-neutral diffusion tendency at
+each tracer point:
+\begin{equation} \label{eq:triad:iso_operator} D_l^T = \frac{1}{b_T}
+ \sum_{\substack{i_p,\,k_p}} \left\{ \delta_{i} \left[{_{i+1/2-i_p}^k
+ {\mathbb{F}_u }_{i_p}^{k_p}} \right] + \delta_{k} \left[
+ {_i^{k+1/2-k_p} {\mathbb{F}_w}_{i_p}^{k_p}} \right] \right\}
+\end{equation}
+where $b_T= e_{1T}\,e_{2T}\,e_{3T}$ is the volume of $T$-cells.
+The diffusion scheme satisfies the following six properties:
+\begin{description}
+\item[$\bullet$ horizontal diffusion] The discretization of the
+ diffusion operator recovers \eqref{eq:triad:lat-normal} the traditional five-point Laplacian in
+ the limit of flat iso-neutral direction :
+ \begin{equation} \label{eq:triad:iso_property0} D_l^T = \frac{1}{b_T} \
+ \delta_{i} \left[ \frac{e_{2u}\,e_{3u}}{e_{1u}} \;
+ \overline\Alts^{\,i} \; \delta_{i+1/2}[T] \right] \qquad
+ \text{when} \quad { _i^k \mathbb{R}_{i_p}^{k_p} }=0
+ \end{equation}
+
+\item[$\bullet$ implicit treatment in the vertical] Only tracer values
+ associated with a single water column appear in the expression
+ \eqref{eq:triad:i33} for the $_{33}$ fluxes, vertical fluxes driven by
+ vertical gradients. This is of paramount importance since it means
+ that a time-implicit algorithm can be used to solve the vertical
+ diffusion equation. This is necessary
+ since the vertical eddy
+ diffusivity associated with this term,
+ \begin{equation}
+ \frac{1}{b_w}\sum_{\substack{i_p, \,k_p}} \left\{
+ {\:}_i^k\mathbb{V}_{i_p}^{k_p} \: \Alts_i^k \: \left(_i^k \mathbb{R}_{i_p}^{k_p}\right)^2
+ \right\} =
+ \frac{1}{4b_w}\sum_{\substack{i_p, \,k_p}} \left\{
+ {b_u}_{i+i_p}^k\: \Alts_i^k \: \left(_i^k \mathbb{R}_{i_p}^{k_p}\right)^2
+ \right\},
+ \end{equation}
+ (where $b_w= e_{1w}\,e_{2w}\,e_{3w}$ is the volume of $w$-cells) can be quite large.
+
+\item[$\bullet$ pure iso-neutral operator] The iso-neutral flux of
+ locally referenced potential density is zero. See
+ \eqref{eq:triad:latflux-rho} and \eqref{eq:triad:vertflux-triad2}.
+
+\item[$\bullet$ conservation of tracer] The iso-neutral diffusion
+ conserves tracer content, $i.e.$
+ \begin{equation} \label{eq:triad:iso_property1} \sum_{i,j,k} \left\{ D_l^T \
+ b_T \right\} = 0
+ \end{equation}
+ This property is trivially satisfied since the iso-neutral diffusive
+ operator is written in flux form.
+
+\item[$\bullet$ no increase of tracer variance] The iso-neutral diffusion
+ does not increase the tracer variance, $i.e.$
+ \begin{equation} \label{eq:triad:iso_property2} \sum_{i,j,k} \left\{ T \ D_l^T
+ \ b_T \right\} \leq 0
+ \end{equation}
+ The property is demonstrated in
+ \S\ref{sec:triad:variance} above. It is a key property for a diffusion
+ term. It means that it is also a dissipation term, $i.e.$ it
+ dissipates the square of the quantity on which it is applied. It
+ therefore ensures that, when the diffusivity coefficient is large
+ enough, the field on which it is applied becomes free of grid-point
+ noise.
+
+\item[$\bullet$ self-adjoint operator] The iso-neutral diffusion
+ operator is self-adjoint, $i.e.$
+ \begin{equation} \label{eq:triad:iso_property3} \sum_{i,j,k} \left\{ S \ D_l^T
+ \ b_T \right\} = \sum_{i,j,k} \left\{ D_l^S \ T \ b_T \right\}
+ \end{equation}
+ In other word, there is no need to develop a specific routine from
+ the adjoint of this operator. We just have to apply the same
+ routine. This property can be demonstrated similarly to the proof of
+ the `no increase of tracer variance' property. The contribution by a
+ single triad towards the left hand side of \eqref{eq:triad:iso_property3}, can
+ be found by replacing $\delta[T]$ by $\delta[S]$ in \eqref{eq:triad:dvar_iso_i}
+ and \eqref{eq:triad:dvar_iso_k}. This results in a term similar to
+ \eqref{eq:triad:perfect-square},
+\begin{equation}
+ \label{eq:triad:TScovar}
+ - \Alts_i^k{\:} _i^k\mathbb{V}_{i_p}^{k_p}
+ \left(
+ \frac{ \delta_{i+ i_p}[T^k] }{ {e_{1u}}_{\,i + i_p}^{\,k} }
+ -{\:}_i^k\mathbb{R}_{i_p}^{k_p}
+ \frac{ \delta_{k+k_p} [T^i] }{{e_{3w}}_{\,i}^{\,k+k_p} }
+ \right)
+ \left(
+ \frac{ \delta_{i+ i_p}[S^k] }{ {e_{1u}}_{\,i + i_p}^{\,k} }
+ -{\:}_i^k\mathbb{R}_{i_p}^{k_p}
+ \frac{ \delta_{k+k_p} [S^i] }{{e_{3w}}_{\,i}^{\,k+k_p} }
+ \right).
+\end{equation}
+This is symmetrical in $T $ and $S$, so exactly the same term arises
+from the discretization of this triad's contribution towards the
+RHS of \eqref{eq:triad:iso_property3}.
+\end{description}
+\subsection{Treatment of the triads at the boundaries}\label{sec:triad:iso_bdry}
+The triad slope can only be defined where both the grid boxes centred at
+the end of the arms exist. Triads that would poke up
+through the upper ocean surface into the atmosphere, or down into the
+ocean floor, must be masked out. See Fig.~\ref{fig:triad:bdry_triads}. Surface layer triads
+$\triad{i}{1}{R}{1/2}{-1/2}$ (magenta) and
+$\triad{i+1}{1}{R}{-1/2}{-1/2}$ (blue) that require density to be
+specified above the ocean surface are masked (Fig.~\ref{fig:triad:bdry_triads}a): this ensures that lateral
+tracer gradients produce no flux through the ocean surface. However,
+to prevent surface noise, it is customary to retain the $_{11}$ contributions towards
+the lateral triad fluxes $\triad[u]{i}{1}{F}{1/2}{-1/2}$ and
+$\triad[u]{i+1}{1}{F}{-1/2}{-1/2}$; this drives diapycnal tracer
+fluxes. Similar comments apply to triads that would intersect the
+ocean floor (Fig.~\ref{fig:triad:bdry_triads}b). Note that both near bottom
+triad slopes $\triad{i}{k}{R}{1/2}{1/2}$ and
+$\triad{i+1}{k}{R}{-1/2}{1/2}$ are masked when either of the $i,k+1$
+or $i+1,k+1$ tracer points is masked, i.e.\ the $i,k+1$ $u$-point is
+masked. The associated lateral fluxes (grey-black dashed line) are
+masked if \np{ln\_botmix\_grif}=false, but left unmasked,
+giving bottom mixing, if \np{ln\_botmix\_grif}=true.
+
+The default option \np{ln\_botmix\_grif}=false is suitable when the
+bbl mixing option is enabled (\key{trabbl}, with \np{nn\_bbl\_ldf}=1),
+or for simple idealized problems. For setups with topography without
+bbl mixing, \np{ln\_botmix\_grif}=true may be necessary.
+% >>>>>>>>>>>>>>>>>>>>>>>>>>>>
+\begin{figure}[h] \begin{center}
+ \includegraphics[width=0.60\textwidth]{./TexFiles/Figures/Fig_GRIFF_bdry_triads}
+ \caption{ \label{fig:triad:bdry_triads}
+ (a) Uppermost model layer $k=1$ with $i,1$ and $i+1,1$ tracer
+ points (black dots), and $i+1/2,1$ $u$-point (blue square). Triad
+ slopes $\triad{i}{1}{R}{1/2}{-1/2}$ (magenta) and $\triad{i+1}{1}{R}{-1/2}{-1/2}$
+ (blue) poking through the ocean surface are masked (faded in
+ figure). However, the lateral $_{11}$ contributions towards
+ $\triad[u]{i}{1}{F}{1/2}{-1/2}$ and $\triad[u]{i+1}{1}{F}{-1/2}{-1/2}$
+ (yellow line) are still applied, giving diapycnal diffusive
+ fluxes.\\
+ (b) Both near bottom triad slopes $\triad{i}{k}{R}{1/2}{1/2}$ and
+ $\triad{i+1}{k}{R}{-1/2}{1/2}$ are masked when either of the $i,k+1$
+ or $i+1,k+1$ tracer points is masked, i.e.\ the $i,k+1$ $u$-point
+ is masked. The associated lateral fluxes (grey-black dashed
+ line) are masked if \np{botmix\_grif}=.false., but left
+ unmasked, giving bottom mixing, if \np{botmix\_grif}=.true.}
+ \end{center} \end{figure}
+% >>>>>>>>>>>>>>>>>>>>>>>>>>>>
+\subsection{ Limiting of the slopes within the interior}\label{sec:triad:limit}
+As discussed in \S\ref{LDF_slp_iso}, iso-neutral slopes relative to
+geopotentials must be bounded everywhere, both for consistency with the small-slope
+approximation and for numerical stability \citep{Cox1987,
+ Griffies_Bk04}. The bound chosen in \NEMO is applied to each
+component of the slope separately and has a value of $1/100$ in the ocean interior.
+%, ramping linearly down above 70~m depth to zero at the surface
+It is of course relevant to the iso-neutral slopes $\tilde{r}_i=r_i+\sigma_i$ relative to
+geopotentials (here the $\sigma_i$ are the slopes of the coordinate surfaces relative to
+geopotentials) \eqref{Eq_PE_slopes_eiv} rather than the slope $r_i$ relative to coordinate
+surfaces, so we require
+\begin{equation*}
+ |\tilde{r}_i|\leq \tilde{r}_\mathrm{max}=0.01.
+\end{equation*}
+and then recalculate the slopes $r_i$ relative to coordinates.
+Each individual triad slope
+ \begin{equation}
+ \label{eq:triad:Rtilde}
+_i^k\tilde{\mathbb{R}}_{i_p}^{k_p} = {}_i^k\mathbb{R}_{i_p}^{k_p} + \frac{\delta_{i+i_p}[z_T^k]}{{e_{1u}}_{\,i + i_p}^{\,k}}
+ \end{equation}
+is limited like this and then the corresponding
+$_i^k\mathbb{R}_{i_p}^{k_p} $ are recalculated and combined to form the fluxes.
+Note that where the slopes have been limited, there is now a non-zero
+iso-neutral density flux that drives dianeutral mixing. In particular this iso-neutral density flux
+is always downwards, and so acts to reduce gravitational potential energy.
+\subsection{Tapering within the surface mixed layer}\label{sec:triad:taper}
+
+Additional tapering of the iso-neutral fluxes is necessary within the
+surface mixed layer. When the Griffies triads are used, we offer two
+options for this.
+\subsubsection{Linear slope tapering within the surface mixed layer}\label{sec:triad:lintaper}
+This is the option activated by the default choice
+\np{ln\_triad\_iso}=false. Slopes $\tilde{r}_i$ relative to
+geopotentials are tapered linearly from their value immediately below the mixed layer to zero at the
+surface, as described in option (c) of Fig.~\ref{Fig_eiv_slp}, to values
+\begin{subequations}
+ \begin{equation}
+ \label{eq:triad:rmtilde}
+ \rMLt =
+ -\frac{z}{h}\left.\tilde{r}_i\right|_{z=-h}\quad \text{ for } z>-h,
+ \end{equation}
+and then the $r_i$ relative to vertical coordinate surfaces are appropriately
+adjusted to
+ \begin{equation}
+ \label{eq:triad:rm}
+ \rML =\rMLt -\sigma_i \quad \text{ for } z>-h.
+ \end{equation}
+\end{subequations}
+Thus the diffusion operator within the mixed layer is given by:
+\begin{equation} \label{eq:triad:iso_tensor_ML}
+D^{lT}=\nabla {\rm {\bf .}}\left( {A^{lT}\;\Re \;\nabla T} \right) \qquad
+\mbox{with}\quad \;\;\Re =\left( {{\begin{array}{*{20}c}
+ 1 \hfill & 0 \hfill & {-\rML[1]}\hfill \\
+ 0 \hfill & 1 \hfill & {-\rML[2]} \hfill \\
+ {-\rML[1]}\hfill & {-\rML[2]} \hfill & {\rML[1]^2+\rML[2]^2} \hfill
+\end{array} }} \right)
+\end{equation}
+
+This slope tapering gives a natural connection between tracer in the
+mixed-layer and in isopycnal layers immediately below, in the
+thermocline. It is consistent with the way the $\tilde{r}_i$ are
+tapered within the mixed layer (see \S\ref{sec:triad:taperskew} below)
+so as to ensure a uniform GM eddy-induced velocity throughout the
+mixed layer. However, it gives a downwards density flux and so acts so
+as to reduce potential energy in the same way as does the slope
+limiting discussed above in \S\ref{sec:triad:limit}.
+
+As in \S\ref{sec:triad:limit} above, the tapering
+\eqref{eq:triad:rmtilde} is applied separately to each triad
+$_i^k\tilde{\mathbb{R}}_{i_p}^{k_p}$, and the
+$_i^k\mathbb{R}_{i_p}^{k_p}$ adjusted. For clarity, we assume
+$z$-coordinates in the following; the conversion from
+$\mathbb{R}$ to $\tilde{\mathbb{R}}$ and back to $\mathbb{R}$ follows exactly as described
+above by \eqref{eq:triad:Rtilde}.
+\begin{enumerate}
+\item Mixed-layer depth is defined so as to avoid including regions of weak
+vertical stratification in the slope definition.
+ At each $i,j$ (simplified to $i$ in
+Fig.~\ref{fig:triad:MLB_triad}), we define the mixed-layer by setting
+the vertical index of the tracer point immediately below the mixed
+layer, $k_\mathrm{ML}$, as the maximum $k$ (shallowest tracer point)
+such that the potential density
+${\rho_0}_{i,k}>{\rho_0}_{i,k_{10}}+\Delta\rho_c$, where $i,k_{10}$ is
+the tracer gridbox within which the depth reaches 10~m. See the left
+side of Fig.~\ref{fig:triad:MLB_triad}. We use the $k_{10}$-gridbox
+instead of the surface gridbox to avoid problems e.g.\ with thin
+daytime mixed-layers. Currently we use the same
+$\Delta\rho_c=0.01\;\mathrm{kg\:m^{-3}}$ for ML triad tapering as is
+used to output the diagnosed mixed-layer depth
+$h_\mathrm{ML}=|z_{W}|_{k_\mathrm{ML}+1/2}$, the depth of the $w$-point
+above the $i,k_\mathrm{ML}$ tracer point.
+
+\item We define `basal' triad slopes
+${\:}_i{\mathbb{R}_\mathrm{base}}_{\,i_p}^{k_p}$ as the slopes
+of those triads whose vertical `arms' go down from the
+$i,k_\mathrm{ML}$ tracer point to the $i,k_\mathrm{ML}-1$ tracer point
+below. This is to ensure that the vertical density gradients
+associated with these basal triad slopes
+${\:}_i{\mathbb{R}_\mathrm{base}}_{\,i_p}^{k_p}$ are
+representative of the thermocline. The four basal triads defined in the bottom part
+of Fig.~\ref{fig:triad:MLB_triad} are then
+\begin{align}
+ {\:}_i{\mathbb{R}_\mathrm{base}}_{\,i_p}^{k_p} &=
+ {\:}^{k_\mathrm{ML}-k_p-1/2}_i{\mathbb{R}_\mathrm{base}}_{\,i_p}^{k_p}, \label{eq:triad:Rbase}
+\\
+\intertext{with e.g.\ the green triad}
+{\:}_i{\mathbb{R}_\mathrm{base}}_{1/2}^{-1/2}&=
+{\:}^{k_\mathrm{ML}}_i{\mathbb{R}_\mathrm{base}}_{\,1/2}^{-1/2}. \notag
+\end{align}
+The vertical flux associated with each of these triads passes through the $w$-point
+$i,k_\mathrm{ML}-1/2$ lying \emph{below} the $i,k_\mathrm{ML}$ tracer point,
+so it is this depth
+\begin{equation}
+ \label{eq:triad:zbase}
+ {z_\mathrm{base}}_{\,i}={z_{w}}_{k_\mathrm{ML}-1/2}
+\end{equation}
+(one gridbox deeper than the
+diagnosed ML depth $z_\mathrm{ML})$ that sets the $h$ used to taper
+the slopes in \eqref{eq:triad:rmtilde}.
+\item Finally, we calculate the adjusted triads
+${\:}_i^k{\mathbb{R}_\mathrm{ML}}_{\,i_p}^{k_p}$ within the mixed
+layer, by multiplying the appropriate
+${\:}_i{\mathbb{R}_\mathrm{base}}_{\,i_p}^{k_p}$ by the ratio of
+the depth of the $w$-point ${z_w}_{k+k_p}$ to ${z_\mathrm{base}}_{\,i}$. For
+instance the green triad centred on $i,k$
+\begin{align}
+ {\:}_i^k{\mathbb{R}_\mathrm{ML}}_{\,1/2}^{-1/2} &=
+\frac{{z_w}_{k-1/2}}{{z_\mathrm{base}}_{\,i}}{\:}_i{\mathbb{R}_\mathrm{base}}_{\,1/2}^{-1/2}
+\notag \\
+\intertext{and more generally}
+ {\:}_i^k{\mathbb{R}_\mathrm{ML}}_{\,i_p}^{k_p} &=
+\frac{{z_w}_{k+k_p}}{{z_\mathrm{base}}_{\,i}}{\:}_i{\mathbb{R}_\mathrm{base}}_{\,i_p}^{k_p}.\label{eq:triad:RML}
+\end{align}
+\end{enumerate}
+
+% >>>>>>>>>>>>>>>>>>>>>>>>>>>>
+\begin{figure}[h]
+ \fcapside {\caption{\label{fig:triad:MLB_triad} Definition of
+ mixed-layer depth and calculation of linearly tapered
+ triads. The figure shows a water column at a given $i,j$
+ (simplified to $i$), with the ocean surface at the top. Tracer points are denoted by
+ bullets, and black lines the edges of the tracer cells; $k$
+ increases upwards. \\
+ \hspace{5 em}We define the mixed-layer by setting the vertical index
+ of the tracer point immediately below the mixed layer,
+ $k_\mathrm{ML}$, as the maximum $k$ (shallowest tracer point)
+ such that ${\rho_0}_{i,k}>{\rho_0}_{i,k_{10}}+\Delta\rho_c$,
+ where $i,k_{10}$ is the tracer gridbox within which the depth
+ reaches 10~m. We calculate the triad slopes within the mixed
+ layer by linearly tapering them from zero (at the surface) to
+ the `basal' slopes, the slopes of the four triads passing through the
+ $w$-point $i,k_\mathrm{ML}-1/2$ (blue square),
+ ${\:}_i{\mathbb{R}_\mathrm{base}}_{\,i_p}^{k_p}$. Triads with
+ different $i_p,k_p$, denoted by different colours, (e.g. the green
+ triad $i_p=1/2,k_p=-1/2$) are tapered to the appropriate basal triad.}}
+ {\includegraphics[width=0.60\textwidth]{./TexFiles/Figures/Fig_GRIFF_MLB_triads}}
+\end{figure}
+% >>>>>>>>>>>>>>>>>>>>>>>>>>>>
+
+\subsubsection{Additional truncation of skew iso-neutral flux
+ components}
+\label{sec:triad:Gerdes-taper}
+The alternative option is activated by setting \np{ln\_triad\_iso} =
+ true. This retains the same tapered slope $\rML$ described above for the
+calculation of the $_{33}$ term of the iso-neutral diffusion tensor (the
+vertical tracer flux driven by vertical tracer gradients), but
+replaces the $\rML$ in the skew term by
+\begin{equation}
+ \label{eq:triad:rm*}
+ \rML^*=\left.\rMLt^2\right/\tilde{r}_i-\sigma_i,
+\end{equation}
+giving a ML diffusive operator
+\begin{equation} \label{eq:triad:iso_tensor_ML2}
+D^{lT}=\nabla {\rm {\bf .}}\left( {A^{lT}\;\Re \;\nabla T} \right) \qquad
+\mbox{with}\quad \;\;\Re =\left( {{\begin{array}{*{20}c}
+ 1 \hfill & 0 \hfill & {-\rML[1]^*}\hfill \\
+ 0 \hfill & 1 \hfill & {-\rML[2]^*} \hfill \\
+ {-\rML[1]^*}\hfill & {-\rML[2]^*} \hfill & {\rML[1]^2+\rML[2]^2} \hfill \\
+\end{array} }} \right).
+\end{equation}
+This operator
+\footnote{To ensure good behaviour where horizontal density
+ gradients are weak, we in fact follow \citet{Gerdes1991} and set
+$\rML^*=\mathrm{sgn}(\tilde{r}_i)\min(|\rMLt^2/\tilde{r}_i|,|\tilde{r}_i|)-\sigma_i$.}
+then has the property it gives no vertical density flux, and so does
+not change the potential energy.
+This approach is similar to multiplying the iso-neutral diffusion
+coefficient by $\tilde{r}_\mathrm{max}^{-2}\tilde{r}_i^{-2}$ for steep
+slopes, as suggested by \citet{Gerdes1991} (see also \citet{Griffies_Bk04}).
+Again it is applied separately to each triad $_i^k\mathbb{R}_{i_p}^{k_p}$
+
+In practice, this approach gives weak vertical tracer fluxes through
+the mixed-layer, as well as vanishing density fluxes. While it is
+theoretically advantageous that it does not change the potential
+energy, it may give a discontinuity between the
+fluxes within the mixed-layer (purely horizontal) and just below (along
+iso-neutral surfaces).
+% This may give strange looking results,
+% particularly where the mixed-layer depth varies strongly laterally.
+% ================================================================
+% Skew flux formulation for Eddy Induced Velocity :
+% ================================================================
+\section{Eddy induced advection formulated as a skew flux}\label{sec:triad:skew-flux}
+
+\subsection{The continuous skew flux formulation}\label{sec:triad:continuous-skew-flux}
+
+ When Gent and McWilliams's [1990] diffusion is used,
+an additional advection term is added. The associated velocity is the so called
+eddy induced velocity, the formulation of which depends on the slopes of iso-
+neutral surfaces. Contrary to the case of iso-neutral mixing, the slopes used
+here are referenced to the geopotential surfaces, $i.e.$ \eqref{Eq_ldfslp_geo}
+is used in $z$-coordinate, and the sum \eqref{Eq_ldfslp_geo}
++ \eqref{Eq_ldfslp_iso} in $z^*$ or $s$-coordinates.
+
+The eddy induced velocity is given by:
+\begin{subequations} \label{eq:triad:eiv}
+\begin{equation}\label{eq:triad:eiv_v}
+\begin{split}
+ u^* & = - \frac{1}{e_{3}}\; \partial_i\psi_1, \\
+ v^* & = - \frac{1}{e_{3}}\; \partial_j\psi_2, \\
+w^* & = \frac{1}{e_{1}e_{2}}\; \left\{ \partial_i \left( e_{2} \, \psi_1\right)
+ + \partial_j \left( e_{1} \, \psi_2\right) \right\},
+\end{split}
+\end{equation}
+where the streamfunctions $\psi_i$ are given by
+\begin{equation} \label{eq:triad:eiv_psi}
+\begin{split}
+\psi_1 & = A_{e} \; \tilde{r}_1, \\
+\psi_2 & = A_{e} \; \tilde{r}_2,
+\end{split}
+\end{equation}
+\end{subequations}
+with $A_{e}$ the eddy induced velocity coefficient, and $\tilde{r}_1$ and $\tilde{r}_2$ the slopes between the iso-neutral and the geopotential surfaces.
+
+The traditional way to implement this additional advection is to add
+it to the Eulerian velocity prior to computing the tracer
+advection. This is implemented if \key{traldf\_eiv} is set in the
+default implementation, where \np{ln\_traldf\_grif} is set
+false. This allows us to take advantage of all the advection schemes
+offered for the tracers (see \S\ref{TRA_adv}) and not just a $2^{nd}$
+order advection scheme. This is particularly useful for passive
+tracers where \emph{positivity} of the advection scheme is of
+paramount importance.
+
+However, when \np{ln\_traldf\_grif} is set true, \NEMO instead
+implements eddy induced advection according to the so-called skew form
+\citep{Griffies_JPO98}. It is based on a transformation of the advective fluxes
+using the non-divergent nature of the eddy induced velocity.
+For example in the (\textbf{i},\textbf{k}) plane, the tracer advective
+fluxes per unit area in $ijk$ space can be
+transformed as follows:
+\begin{flalign*}
+\begin{split}
+\textbf{F}_\mathrm{eiv}^T =
+\begin{pmatrix}
+ {e_{2}\,e_{3}\; u^*} \\
+ {e_{1}\,e_{2}\; w^*} \\
+\end{pmatrix} \; T
+&=
+\begin{pmatrix}
+ { - \partial_k \left( e_{2} \,\psi_1 \right) \; T \;} \\
+ {+ \partial_i \left( e_{2} \, \psi_1 \right) \; T \;} \\
+\end{pmatrix} \\
+&=
+\begin{pmatrix}
+ { - \partial_k \left( e_{2} \, \psi_1 \; T \right) \;} \\
+ {+ \partial_i \left( e_{2} \,\psi_1 \; T \right) \;} \\
+\end{pmatrix}
+ +
+\begin{pmatrix}
+ {+ e_{2} \, \psi_1 \; \partial_k T} \\
+ { - e_{2} \, \psi_1 \; \partial_i T} \\
+\end{pmatrix}
+\end{split}
+\end{flalign*}
+and since the eddy induced velocity field is non-divergent, we end up with the skew
+form of the eddy induced advective fluxes per unit area in $ijk$ space:
+\begin{equation} \label{eq:triad:eiv_skew_ijk}
+\textbf{F}_\mathrm{eiv}^T = \begin{pmatrix}
+ {+ e_{2} \, \psi_1 \; \partial_k T} \\
+ { - e_{2} \, \psi_1 \; \partial_i T} \\
+ \end{pmatrix}
+\end{equation}
+The total fluxes per unit physical area are then
+\begin{equation}\label{eq:triad:eiv_skew_physical}
+\begin{split}
+ f^*_1 & = \frac{1}{e_{3}}\; \psi_1 \partial_k T \\
+ f^*_2 & = \frac{1}{e_{3}}\; \psi_2 \partial_k T \\
+ f^*_3 & = -\frac{1}{e_{1}e_{2}}\; \left\{ e_{2} \psi_1 \partial_i T
+ + e_{1} \psi_2 \partial_j T \right\}. \\
+\end{split}
+\end{equation}
+Note that Eq.~ \eqref{eq:triad:eiv_skew_physical} takes the same form whatever the
+vertical coordinate, though of course the slopes
+$\tilde{r}_i$ which define the $\psi_i$ in \eqref{eq:triad:eiv_psi} are relative to geopotentials.
+The tendency associated with eddy induced velocity is then simply the convergence
+of the fluxes (\ref{eq:triad:eiv_skew_ijk}, \ref{eq:triad:eiv_skew_physical}), so
+\begin{equation} \label{eq:triad:skew_eiv_conv}
+\frac{\partial T}{\partial t}= -\frac{1}{e_1 \, e_2 \, e_3 } \left[
+ \frac{\partial}{\partial i} \left( e_2 \psi_1 \partial_k T\right)
+ + \frac{\partial}{\partial j} \left( e_1 \;
+ \psi_2 \partial_k T\right)
+ - \frac{\partial}{\partial k} \left( e_{2} \psi_1 \partial_i T
+ + e_{1} \psi_2 \partial_j T \right) \right]
+\end{equation}
+ It naturally conserves the tracer content, as it is expressed in flux
+ form. Since it has the same divergence as the advective form it also
+ preserves the tracer variance.
+
+\subsection{The discrete skew flux formulation}
+The skew fluxes in (\ref{eq:triad:eiv_skew_physical}, \ref{eq:triad:eiv_skew_ijk}), like the off-diagonal terms
+(\ref{eq:triad:i13c}, \ref{eq:triad:i31c}) of the small angle diffusion tensor, are best
+expressed in terms of the triad slopes, as in Fig.~\ref{fig:triad:ISO_triad}
+and Eqs~(\ref{eq:triad:i13}, \ref{eq:triad:i31}); but now in terms of the triad slopes
+$\tilde{\mathbb{R}}$ relative to geopotentials instead of the
+$\mathbb{R}$ relative to coordinate surfaces. The discrete form of
+\eqref{eq:triad:eiv_skew_ijk} using the slopes \eqref{eq:triad:R} and
+defining $A_e$ at $T$-points is then given by:
+
+
+\begin{subequations}\label{eq:triad:allskewflux}
+ \begin{flalign}\label{eq:triad:vect_skew_flux}
+ \vect{F}_\mathrm{eiv}(T) &\equiv
+ \sum_{\substack{i_p,\,k_p}}
+ \begin{pmatrix}
+ {_{i+1/2-i_p}^k {\mathbb{S}_u}_{i_p}^{k_p} } (T) \\
+ \\
+ {_i^{k+1/2-k_p} {\mathbb{S}_w}_{i_p}^{k_p} } (T) \\
+ \end{pmatrix},
+ \end{flalign}
+ where the skew flux in the $i$-direction associated with a given
+ triad is (\ref{eq:triad:latflux-triad}, \ref{eq:triad:triadfluxu}):
+ \begin{align}
+ \label{eq:triad:skewfluxu}
+ _i^k {\mathbb{S}_u}_{i_p}^{k_p} (T) &= + \quarter {A_e}_i^k{
+ \:}\frac{{b_u}_{i+i_p}^k}{{e_{1u}}_{\,i + i_p}^{\,k}}
+ \ {_i^k\tilde{\mathbb{R}}_{i_p}^{k_p}} \
+ \frac{ \delta_{k+k_p} [T^i] }{{e_{3w}}_{\,i}^{\,k+k_p} },
+ \\
+ \intertext{and \eqref{eq:triad:triadfluxw} in the $k$-direction, changing the sign
+ to be consistent with \eqref{eq:triad:eiv_skew_ijk}:}
+ _i^k {\mathbb{S}_w}_{i_p}^{k_p} (T)
+ &= -\quarter {A_e}_i^k{\: }\frac{{b_u}_{i+i_p}^k}{{e_{3w}}_{\,i}^{\,k+k_p}}
+ {_i^k\tilde{\mathbb{R}}_{i_p}^{k_p}}\frac{ \delta_{i+ i_p}[T^k] }{ {e_{1u}}_{\,i + i_p}^{\,k} }.\label{eq:triad:skewfluxw}
+ \end{align}
+\end{subequations}
+
+Such a discretisation is consistent with the iso-neutral
+operator as it uses the same definition for the slopes. It also
+ensures the following two key properties.
+\subsubsection{No change in tracer variance}
+The discretization conserves tracer variance, $i.e.$ it does not
+include a diffusive component but is a `pure' advection term. This can
+be seen
+%either from Appendix \ref{Apdx_eiv_skew} or
+by considering the
+fluxes associated with a given triad slope
+$_i^k{\mathbb{R}}_{i_p}^{k_p} (T)$. For, following
+\S\ref{sec:triad:variance} and \eqref{eq:triad:dvar_iso_i}, the
+associated horizontal skew-flux $_i^k{\mathbb{S}_u}_{i_p}^{k_p} (T)$
+drives a net rate of change of variance, summed over the two
+$T$-points $i+i_p-\half,k$ and $i+i_p+\half,k$, of
+\begin{equation}
+\label{eq:triad:dvar_eiv_i}
+ _i^k{\mathbb{S}_u}_{i_p}^{k_p} (T)\,\delta_{i+ i_p}[T^k],
+\end{equation}
+while the associated vertical skew-flux gives a variance change summed over the
+$T$-points $i,k+k_p-\half$ (above) and $i,k+k_p+\half$ (below) of
+\begin{equation}
+\label{eq:triad:dvar_eiv_k}
+ _i^k{\mathbb{S}_w}_{i_p}^{k_p} (T) \,\delta_{k+ k_p}[T^i].
+\end{equation}
+Inspection of the definitions (\ref{eq:triad:skewfluxu}, \ref{eq:triad:skewfluxw})
+shows that these two variance changes (\ref{eq:triad:dvar_eiv_i}, \ref{eq:triad:dvar_eiv_k})
+sum to zero. Hence the two fluxes associated with each triad make no
+net contribution to the variance budget.
+
+\subsubsection{Reduction in gravitational PE}
+The vertical density flux associated with the vertical skew-flux
+always has the same sign as the vertical density gradient; thus, so
+long as the fluid is stable (the vertical density gradient is
+negative) the vertical density flux is negative (downward) and hence
+reduces the gravitational PE.
+
+For the change in gravitational PE driven by the $k$-flux is
+\begin{align}
+ \label{eq:triad:vert_densityPE}
+ g {e_{3w}}_{\,i}^{\,k+k_p}{\mathbb{S}_w}_{i_p}^{k_p} (\rho)
+ &=g {e_{3w}}_{\,i}^{\,k+k_p}\left[-\alpha _i^k {\:}_i^k
+ {\mathbb{S}_w}_{i_p}^{k_p} (T) + \beta_i^k {\:}_i^k
+ {\mathbb{S}_w}_{i_p}^{k_p} (S) \right]. \notag \\
+\intertext{Substituting ${\:}_i^k {\mathbb{S}_w}_{i_p}^{k_p}$ from
+ \eqref{eq:triad:skewfluxw}, gives}
+% and separating out
+% $\rtriadt{R}=\rtriad{R} + \delta_{i+i_p}[z_T^k]$,
+% gives two terms. The
+% first $\rtriad{R}$ term (the only term for $z$-coordinates) is:
+ &=-\quarter g{A_e}_i^k{\: }{b_u}_{i+i_p}^k {_i^k\tilde{\mathbb{R}}_{i_p}^{k_p}}
+\frac{ -\alpha _i^k\delta_{i+ i_p}[T^k]+ \beta_i^k\delta_{i+ i_p}[S^k]} { {e_{1u}}_{\,i + i_p}^{\,k} } \notag \\
+ &=+\quarter g{A_e}_i^k{\: }{b_u}_{i+i_p}^k
+ \left({_i^k\mathbb{R}_{i_p}^{k_p}}+\frac{\delta_{i+i_p}[z_T^k]}{{e_{1u}}_{\,i + i_p}^{\,k}}\right) {_i^k\mathbb{R}_{i_p}^{k_p}}
+\frac{-\alpha_i^k \delta_{k+ k_p}[T^i]+ \beta_i^k\delta_{k+ k_p}[S^i]} {{e_{3w}}_{\,i}^{\,k+k_p}},
+\end{align}
+using the definition of the triad slope $\rtriad{R}$,
+\eqref{eq:triad:R} to express $-\alpha _i^k\delta_{i+ i_p}[T^k]+
+\beta_i^k\delta_{i+ i_p}[S^k]$ in terms of $-\alpha_i^k \delta_{k+
+ k_p}[T^i]+ \beta_i^k\delta_{k+ k_p}[S^i]$.
+
+Where the coordinates slope, the $i$-flux gives a PE change
+\begin{multline}
+ \label{eq:triad:lat_densityPE}
+ g \delta_{i+i_p}[z_T^k]
+\left[
+-\alpha _i^k {\:}_i^k {\mathbb{S}_u}_{i_p}^{k_p} (T) + \beta_i^k {\:}_i^k {\mathbb{S}_u}_{i_p}^{k_p} (S)
+\right] \\
+= +\quarter g{A_e}_i^k{\: }{b_u}_{i+i_p}^k
+ \frac{\delta_{i+i_p}[z_T^k]}{{e_{1u}}_{\,i + i_p}^{\,k}}
+\left({_i^k\mathbb{R}_{i_p}^{k_p}}+\frac{\delta_{i+i_p}[z_T^k]}{{e_{1u}}_{\,i + i_p}^{\,k}}\right)
+\frac{-\alpha_i^k \delta_{k+ k_p}[T^i]+ \beta_i^k\delta_{k+ k_p}[S^i]} {{e_{3w}}_{\,i}^{\,k+k_p}},
+\end{multline}
+(using \eqref{eq:triad:skewfluxu}) and so the total PE change
+\eqref{eq:triad:vert_densityPE} + \eqref{eq:triad:lat_densityPE} associated with the triad fluxes is
+\begin{multline}
+ \label{eq:triad:tot_densityPE}
+ g{e_{3w}}_{\,i}^{\,k+k_p}{\mathbb{S}_w}_{i_p}^{k_p} (\rho) +
+g\delta_{i+i_p}[z_T^k] {\:}_i^k {\mathbb{S}_u}_{i_p}^{k_p} (\rho) \\
+= +\quarter g{A_e}_i^k{\: }{b_u}_{i+i_p}^k
+ \left({_i^k\mathbb{R}_{i_p}^{k_p}}+\frac{\delta_{i+i_p}[z_T^k]}{{e_{1u}}_{\,i + i_p}^{\,k}}\right)^2
+\frac{-\alpha_i^k \delta_{k+ k_p}[T^i]+ \beta_i^k\delta_{k+ k_p}[S^i]} {{e_{3w}}_{\,i}^{\,k+k_p}}.
+\end{multline}
+Where the fluid is stable, with $-\alpha_i^k \delta_{k+ k_p}[T^i]+
+\beta_i^k\delta_{k+ k_p}[S^i]<0$, this PE change is negative.
+
+\subsection{Treatment of the triads at the boundaries}\label{sec:triad:skew_bdry}
+Triad slopes \rtriadt{R} used for the calculation of the eddy-induced skew-fluxes
+are masked at the boundaries in exactly the same way as are the triad
+slopes \rtriad{R} used for the iso-neutral diffusive fluxes, as
+described in \S\ref{sec:triad:iso_bdry} and
+Fig.~\ref{fig:triad:bdry_triads}. Thus surface layer triads
+$\triadt{i}{1}{R}{1/2}{-1/2}$ and $\triadt{i+1}{1}{R}{-1/2}{-1/2}$ are
+masked, and both near bottom triad slopes $\triadt{i}{k}{R}{1/2}{1/2}$
+and $\triadt{i+1}{k}{R}{-1/2}{1/2}$ are masked when either of the
+$i,k+1$ or $i+1,k+1$ tracer points is masked, i.e.\ the $i,k+1$
+$u$-point is masked. The namelist parameter \np{ln\_botmix\_grif} has
+no effect on the eddy-induced skew-fluxes.
+
+\subsection{ Limiting of the slopes within the interior}\label{sec:triad:limitskew}
+Presently, the iso-neutral slopes $\tilde{r}_i$ relative
+to geopotentials are limited to be less than $1/100$, exactly as in
+calculating the iso-neutral diffusion, \S \ref{sec:triad:limit}. Each
+individual triad \rtriadt{R} is so limited.
+
+\subsection{Tapering within the surface mixed layer}\label{sec:triad:taperskew}
+The slopes $\tilde{r}_i$ relative to
+geopotentials (and thus the individual triads \rtriadt{R}) are always tapered linearly from their value immediately below the mixed layer to zero at the
+surface \eqref{eq:triad:rmtilde}, as described in \S\ref{sec:triad:lintaper}. This is
+option (c) of Fig.~\ref{Fig_eiv_slp}. This linear tapering for the
+slopes used to calculate the eddy-induced fluxes is
+unaffected by the value of \np{ln\_triad\_iso}.
+
+The justification for this linear slope tapering is that, for $A_e$
+that is constant or varies only in the horizontal (the most commonly
+used options in \NEMO: see \S\ref{LDF_coef}), it is
+equivalent to a horizontal eiv (eddy-induced velocity) that is uniform
+within the mixed layer \eqref{eq:triad:eiv_v}. This ensures that the
+eiv velocities do not restratify the mixed layer \citep{Treguier1997,
+ Danabasoglu_al_2008}. Equivantly, in terms
+of the skew-flux formulation we use here, the
+linear slope tapering within the mixed-layer gives a linearly varying
+vertical flux, and so a tracer convergence uniform in depth (the
+horizontal flux convergence is relatively insignificant within the mixed-layer).
+
+\subsection{Streamfunction diagnostics}\label{sec:triad:sfdiag}
+Where the namelist parameter \np{ln\_traldf\_gdia}=true, diagnosed
+mean eddy-induced velocities are output. Each time step,
+streamfunctions are calculated in the $i$-$k$ and $j$-$k$ planes at
+$uw$ (integer +1/2 $i$, integer $j$, integer +1/2 $k$) and $vw$
+(integer $i$, integer +1/2 $j$, integer +1/2 $k$) points (see Table
+\ref{Tab_cell}) respectively. We follow \citep{Griffies_Bk04} and
+calculate the streamfunction at a given $uw$-point from the
+surrounding four triads according to:
+\begin{equation}
+ \label{eq:triad:sfdiagi}
+ {\psi_1}_{i+1/2}^{k+1/2}={\quarter}\sum_{\substack{i_p,\,k_p}}
+ {A_e}_{i+1/2-i_p}^{k+1/2-k_p}\:\triadd{i+1/2-i_p}{k+1/2-k_p}{R}{i_p}{k_p}.
+\end{equation}
+The streamfunction $\psi_1$ is calculated similarly at $vw$ points.
+The eddy-induced velocities are then calculated from the
+straightforward discretisation of \eqref{eq:triad:eiv_v}:
+\begin{equation}\label{eq:triad:eiv_v_discrete}
+\begin{split}
+ {u^*}_{i+1/2}^{k} & = - \frac{1}{{e_{3u}}_{i}^{k}}\left({\psi_1}_{i+1/2}^{k+1/2}-{\psi_1}_{i+1/2}^{k+1/2}\right), \\
+ {v^*}_{j+1/2}^{k} & = - \frac{1}{{e_{3v}}_{j}^{k}}\left({\psi_2}_{j+1/2}^{k+1/2}-{\psi_2}_{j+1/2}^{k+1/2}\right), \\
+ {w^*}_{i,j}^{k+1/2} & = \frac{1}{e_{1t}e_{2t}}\; \left\{
+ {e_{2u}}_{i+1/2}^{k+1/2} \,{\psi_1}_{i+1/2}^{k+1/2} -
+ {e_{2u}}_{i-1/2}^{k+1/2} \,{\psi_1}_{i-1/2}^{k+1/2} \right. + \\
+\phantom{=} & \qquad\qquad\left. {e_{2v}}_{j+1/2}^{k+1/2} \,{\psi_2}_{j+1/2}^{k+1/2} - {e_{2v}}_{j-1/2}^{k+1/2} \,{\psi_2}_{j-1/2}^{k+1/2} \right\},
+\end{split}
+\end{equation}
Index: /branches/UKMO/dev_r5518_GO6_under_ice_relax_dr_hook/DOC/TexFiles/Chapters/Chap_ASM.tex
===================================================================
--- /branches/UKMO/dev_r5518_GO6_under_ice_relax_dr_hook/DOC/TexFiles/Chapters/Chap_ASM.tex (revision 11643)
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@@ -0,0 +1,173 @@
+% ================================================================
+% Chapter Assimilation increments (ASM)
+% ================================================================
+\chapter{Apply assimilation increments (ASM)}
+\label{ASM}
+
+Authors: D. Lea, M. Martin, K. Mogensen, A. Weaver, ... % do we keep
+
+\minitoc
+
+
+\newpage
+$\ $\newline % force a new line
+
+The ASM code adds the functionality to apply increments to the model variables:
+temperature, salinity, sea surface height, velocity and sea ice concentration.
+These are read into the model from a NetCDF file which may be produced by separate data
+assimilation code. The code can also output model background fields which are used
+as an input to data assimilation code. This is all controlled by the namelist
+\textit{\ngn{nam\_asminc} }. There is a brief description of all the namelist options
+provided. To build the ASM code \key{asminc} must be set.
+
+%===============================================================
+
+\section{Direct initialization}
+\label{ASM_DI}
+
+Direct initialization (DI) refers to the instantaneous correction
+of the model background state using the analysis increment.
+DI is used when \np{ln\_asmdin} is set to true.
+
+\section{Incremental Analysis Updates}
+\label{ASM_IAU}
+
+Rather than updating the model state directly with the analysis increment,
+it may be preferable to introduce the increment gradually into the ocean
+model in order to minimize spurious adjustment processes. This technique
+is referred to as Incremental Analysis Updates (IAU) \citep{Bloom_al_MWR96}.
+IAU is a common technique used with 3D assimilation methods such as 3D-Var or OI.
+IAU is used when \np{ln\_asmiau} is set to true.
+
+With IAU, the model state trajectory ${\bf x}$ in the assimilation window
+($t_{0} \leq t_{i} \leq t_{N}$)
+is corrected by adding the analysis increments for temperature, salinity, horizontal velocity and SSH
+as additional tendency terms to the prognostic equations:
+\begin{eqnarray} \label{eq:wa_traj_iau}
+{\bf x}^{a}(t_{i}) = M(t_{i}, t_{0})[{\bf x}^{b}(t_{0})]
+\; + \; F_{i} \delta \tilde{\bf x}^{a}
+\end{eqnarray}
+where $F_{i}$ is a weighting function for applying the increments $\delta
+\tilde{\bf x}^{a}$ defined such that $\sum_{i=1}^{N} F_{i}=1$.
+${\bf x}^b$ denotes the model initial state and ${\bf x}^a$ is the model state
+after the increments are applied.
+To control the adjustment time of the model to the increment,
+the increment can be applied over an arbitrary sub-window,
+$t_{m} \leq t_{i} \leq t_{n}$, of the main assimilation window,
+where $t_{0} \leq t_{m} \leq t_{i}$ and $t_{i} \leq t_{n} \leq t_{N}$,
+Typically the increments are spread evenly over the full window.
+In addition, two different weighting functions have been implemented.
+The first function employs constant weights,
+\begin{eqnarray} \label{eq:F1_i}
+F^{(1)}_{i}
+=\left\{ \begin{array}{ll}
+ 0 & {\rm if} \; \; \; t_{i} < t_{m} \\
+ 1/M & {\rm if} \; \; \; t_{m} < t_{i} \leq t_{n} \\
+ 0 & {\rm if} \; \; \; t_{i} > t_{n}
+ \end{array} \right.
+\end{eqnarray}
+where $M = m-n$.
+The second function employs peaked hat-like weights in order to give maximum
+weight in the centre of the sub-window, with the weighting reduced
+linearly to a small value at the window end-points:
+\begin{eqnarray} \label{eq:F2_i}
+F^{(2)}_{i}
+=\left\{ \begin{array}{ll}
+ 0 & {\rm if} \; \; \; t_{i} < t_{m} \\
+ \alpha \, i & {\rm if} \; \; \; t_{m} \leq t_{i} \leq t_{M/2} \\
+ \alpha \, (M - i +1) & {\rm if} \; \; \; t_{M/2} < t_{i} \leq t_{n} \\
+ 0 & {\rm if} \; \; \; t_{i} > t_{n}
+ \end{array} \right.
+\end{eqnarray}
+where $\alpha^{-1} = \sum_{i=1}^{M/2} 2i$ and $M$ is assumed to be even.
+The weights described by \eqref{eq:F2_i} provide a
+smoother transition of the analysis trajectory from one assimilation cycle
+to the next than that described by \eqref{eq:F1_i}.
+
+%==========================================================================
+% Divergence damping description %%%
+\section{Divergence damping initialisation}
+\label{ASM_details}
+
+The velocity increments may be initialized by the iterative application of
+a divergence damping operator. In iteration step $n$ new estimates of
+velocity increments $u^{n}_I$ and $v^{n}_I$ are updated by:
+\begin{equation} \label{eq:asm_dmp}
+\left\{ \begin{aligned}
+ u^{n}_I = u^{n-1}_I + \frac{1}{e_{1u} } \delta _{i+1/2} \left( {A_D
+\;\chi^{n-1}_I } \right) \\
+\\
+ v^{n}_I = v^{n-1}_I + \frac{1}{e_{2v} } \delta _{j+1/2} \left( {A_D
+\;\chi^{n-1}_I } \right) \\
+\end{aligned} \right.,
+\end{equation}
+where
+\begin{equation} \label{eq:asm_div}
+\chi^{n-1}_I = \frac{1}{e_{1t}\,e_{2t}\,e_{3t} }
+ \left( {\delta _i \left[ {e_{2u}\,e_{3u}\,u^{n-1}_I} \right]
+ +\delta _j \left[ {e_{1v}\,e_{3v}\,v^{n-1}_I} \right]} \right).
+\end{equation}
+By the application of \eqref{eq:asm_dmp} and \eqref{eq:asm_dmp} the divergence is filtered
+in each iteration, and the vorticity is left unchanged. In the presence of coastal boundaries
+with zero velocity increments perpendicular to the coast the divergence is strongly damped.
+This type of the initialisation reduces the vertical velocity magnitude and alleviates the
+problem of the excessive unphysical vertical mixing in the first steps of the model
+integration \citep{Talagrand_JAS72, Dobricic_al_OS07}. Diffusion coefficients are defined as
+$A_D = \alpha e_{1t} e_{2t}$, where $\alpha = 0.2$. The divergence damping is activated by
+assigning to \np{nn\_divdmp} in the \textit{nam\_asminc} namelist a value greater than zero.
+By choosing this value to be of the order of 100 the increments in the vertical velocity will
+be significantly reduced.
+
+
+%==========================================================================
+
+\section{Implementation details}
+\label{ASM_details}
+
+Here we show an example \ngn{namasm} namelist and the header of an example assimilation
+increments file on the ORCA2 grid.
+
+%------------------------------------------namasm-----------------------------------------------------
+\namdisplay{namasm}
+%-------------------------------------------------------------------------------------------------------------
+
+The header of an assimilation increments file produced using the NetCDF tool
+\mbox{\textit{ncdump~-h}} is shown below
+
+\begin{alltt}
+\tiny
+\begin{verbatim}
+netcdf assim_background_increments {
+dimensions:
+ x = 182 ;
+ y = 149 ;
+ z = 31 ;
+ t = UNLIMITED ; // (1 currently)
+variables:
+ float nav_lon(y, x) ;
+ float nav_lat(y, x) ;
+ float nav_lev(z) ;
+ double time_counter(t) ;
+ double time ;
+ double z_inc_dateb ;
+ double z_inc_datef ;
+ double bckint(t, z, y, x) ;
+ double bckins(t, z, y, x) ;
+ double bckinu(t, z, y, x) ;
+ double bckinv(t, z, y, x) ;
+ double bckineta(t, y, x) ;
+
+// global attributes:
+ :DOMAIN_number_total = 1 ;
+ :DOMAIN_number = 0 ;
+ :DOMAIN_dimensions_ids = 1, 2 ;
+ :DOMAIN_size_global = 182, 149 ;
+ :DOMAIN_size_local = 182, 149 ;
+ :DOMAIN_position_first = 1, 1 ;
+ :DOMAIN_position_last = 182, 149 ;
+ :DOMAIN_halo_size_start = 0, 0 ;
+ :DOMAIN_halo_size_end = 0, 0 ;
+ :DOMAIN_type = "BOX" ;
+}
+\end{verbatim}
+\end{alltt}
Index: /branches/UKMO/dev_r5518_GO6_under_ice_relax_dr_hook/DOC/TexFiles/Chapters/Chap_CFG.tex
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@@ -0,0 +1,312 @@
+% ================================================================
+% Chapter � Configurations
+% ================================================================
+\chapter{Configurations}
+\label{CFG}
+\minitoc
+
+\newpage
+$\ $\newline % force a new ligne
+
+% ================================================================
+% Introduction
+% ================================================================
+\section{Introduction}
+\label{CFG_intro}
+
+
+The purpose of this part of the manual is to introduce the \NEMO reference configurations.
+These configurations are offered as means to explore various numerical and physical options,
+thus allowing the user to verify that the code is performing in a manner consistent with that
+we are running. This form of verification is critical as one adopts the code for his or her particular
+research purposes. The test cases also provide a sense for some of the options available
+in the code, though by no means are all options exercised in the reference configurations.
+
+Configuration is defined mainly through the \ngn{namcfg} namelist variables:
+%------------------------------------------namcfg----------------------------------------------------
+\namdisplay{namcfg}
+%-------------------------------------------------------------------------------------------------------------
+
+% ================================================================
+% 1D model configuration
+% ================================================================
+\section{Water column model: 1D model (C1D) (\key{c1d}) }
+\label{CFG_c1d}
+
+The 1D model option simulates a stand alone water column within the 3D \NEMO system.
+It can be applied to the ocean alone or to the ocean-ice system and can include passive tracers
+or a biogeochemical model. It is set up by defining the position of the 1D water column in the grid
+(see \textit{CONFIG/SHARED/namelist\_ref} ).
+The 1D model is a very useful tool
+\textit{(a)} to learn about the physics and numerical treatment of vertical mixing processes ;
+\textit{(b)} to investigate suitable parameterisations of unresolved turbulence (surface wave
+breaking, Langmuir circulation, ...) ;
+\textit{(c)} to compare the behaviour of different vertical mixing schemes ;
+\textit{(d)} to perform sensitivity studies on the vertical diffusion at a particular point of an ocean domain ;
+\textit{(d)} to produce extra diagnostics, without the large memory requirement of the full 3D model.
+
+The methodology is based on the use of the zoom functionality over the smallest possible
+domain : a 3x3 domain centered on the grid point of interest,
+with some extra routines. There is no need to define a new mesh, bathymetry,
+initial state or forcing, since the 1D model will use those of the configuration it is a zoom of.
+The chosen grid point is set in \textit{\ngn{namcfg}} namelist by setting the \np{jpizoom} and \np{jpjzoom}
+parameters to the indices of the location of the chosen grid point.
+
+The 1D model has some specifies. First, all the horizontal derivatives are assumed to be zero, and
+second, the two components of the velocity are moved on a $T$-point.
+Therefore, defining \key{c1d} changes five main things in the code behaviour:
+\begin{description}
+\item[(1)] the lateral boundary condition routine (\rou{lbc\_lnk}) set the value of the central column
+of the 3x3 domain is imposed over the whole domain ;
+\item[(3)] a call to \rou{lbc\_lnk} is systematically done when reading input data ($i.e.$ in \mdl{iom}) ;
+\item[(3)] a simplified \rou{stp} routine is used (\rou{stp\_c1d}, see \mdl{step\_c1d} module) in which
+both lateral tendancy terms and lateral physics are not called ;
+\item[(4)] the vertical velocity is zero (so far, no attempt at introducing a Ekman pumping velocity
+has been made) ;
+\item[(5)] a simplified treatment of the Coriolis term is performed as $U$- and $V$-points are the same
+(see \mdl{dyncor\_c1d}).
+\end{description}
+All the relevant \textit{\_c1d} modules can be found in the NEMOGCM/NEMO/OPA\_SRC/C1D directory of
+the \NEMO distribution.
+
+% to be added: a test case on the yearlong Ocean Weather Station (OWS) Papa dataset of Martin (1985)
+
+% ================================================================
+% ORCA family configurations
+% ================================================================
+\section{ORCA family: global ocean with tripolar grid }
+\label{CFG_orca}
+
+The ORCA family is a series of global ocean configurations that are run together with
+the LIM sea-ice model (ORCA-LIM) and possibly with PISCES biogeochemical model
+(ORCA-LIM-PISCES), using various resolutions.
+The appropriate \textit{\&namcfg} namelist is available in \textit{CONFIG/ORCA2\_LIM/EXP00/namelist\_cfg}
+for ORCA2 and in \textit{CONFIG/SHARED/README\_other\_configurations\_namelist\_namcfg}
+for other resolutions
+
+
+%>>>>>>>>>>>>>>>>>>>>>>>>>>>>
+\begin{figure}[!t] \begin{center}
+\includegraphics[width=0.98\textwidth]{./TexFiles/Figures/Fig_ORCA_NH_mesh.pdf}
+\caption{ \label{Fig_MISC_ORCA_msh}
+ORCA mesh conception. The departure from an isotropic Mercator grid start poleward of 20\deg N.
+The two "north pole" are the foci of a series of embedded ellipses (blue curves)
+which are determined analytically and form the i-lines of the ORCA mesh (pseudo latitudes).
+Then, following \citet{Madec_Imbard_CD96}, the normal to the series of ellipses (red curves) is computed
+which provide the j-lines of the mesh (pseudo longitudes). }
+\end{center} \end{figure}
+%>>>>>>>>>>>>>>>>>>>>>>>>>>>>
+
+% -------------------------------------------------------------------------------------------------------------
+% ORCA tripolar grid
+% -------------------------------------------------------------------------------------------------------------
+\subsection{ORCA tripolar grid}
+\label{CFG_orca_grid}
+
+The ORCA grid is a tripolar is based on the semi-analytical method of \citet{Madec_Imbard_CD96}.
+It allows to construct a global orthogonal curvilinear ocean mesh which has no singularity point inside
+the computational domain since two north mesh poles are introduced and placed on lands.
+The method involves defining an analytical set of mesh parallels in the stereographic polar plan,
+computing the associated set of mesh meridians, and projecting the resulting mesh onto the sphere.
+The set of mesh parallels used is a series of embedded ellipses which foci are the two mesh north
+poles (Fig.~\ref{Fig_MISC_ORCA_msh}). The resulting mesh presents no loss of continuity in
+either the mesh lines or the scale factors, or even the scale factor derivatives over the whole
+ocean domain, as the mesh is not a composite mesh.
+%>>>>>>>>>>>>>>>>>>>>>>>>>>>>
+\begin{figure}[!tbp] \begin{center}
+\includegraphics[width=1.0\textwidth]{./TexFiles/Figures/Fig_ORCA_NH_msh05_e1_e2.pdf}
+\includegraphics[width=0.80\textwidth]{./TexFiles/Figures/Fig_ORCA_aniso.pdf}
+\caption { \label{Fig_MISC_ORCA_e1e2}
+\textit{Top}: Horizontal scale factors ($e_1$, $e_2$) and
+\textit{Bottom}: ratio of anisotropy ($e_1 / e_2$)
+for ORCA 0.5\deg ~mesh. South of 20\deg N a Mercator grid is used ($e_1 = e_2$)
+so that the anisotropy ratio is 1. Poleward of 20\deg N, the two "north pole"
+introduce a weak anisotropy over the ocean areas ($< 1.2$) except in vicinity of Victoria Island
+(Canadian Arctic Archipelago). }
+\end{center} \end{figure}
+%>>>>>>>>>>>>>>>>>>>>>>>>>>>>
+
+
+The method is applied to Mercator grid ($i.e.$ same zonal and meridional grid spacing) poleward
+of $20\deg$N, so that the Equator is a mesh line, which provides a better numerical solution
+for equatorial dynamics. The choice of the series of embedded ellipses (position of the foci and
+variation of the ellipses) is a compromise between maintaining the ratio of mesh anisotropy
+($e_1 / e_2$) close to one in the ocean (especially in area of strong eddy activities such as
+the Gulf Stream) and keeping the smallest scale factor in the northern hemisphere larger
+than the smallest one in the southern hemisphere.
+The resulting mesh is shown in Fig.~\ref{Fig_MISC_ORCA_msh} and \ref{Fig_MISC_ORCA_e1e2}
+for a half a degree grid (ORCA\_R05). The smallest ocean scale factor is found in along
+Antarctica, while the ratio of anisotropy remains close to one except near the Victoria Island
+in the Canadian Archipelago.
+
+% -------------------------------------------------------------------------------------------------------------
+% ORCA-LIM(-PISCES) configurations
+% -------------------------------------------------------------------------------------------------------------
+\subsection{ORCA pre-defined resolution}
+\label{CFG_orca_resolution}
+
+
+The NEMO system is provided with five built-in ORCA configurations which differ in the
+horizontal resolution. The value of the resolution is given by the resolution at the Equator
+expressed in degrees. Each of configuration is set through the \textit{\ngn{namcfg}} namelist,
+which sets the grid size and configuration
+name parameters (Tab. \ref{Tab_ORCA}).
+.
+
+%--------------------------------------------------TABLE--------------------------------------------------
+\begin{table}[!t] \begin{center}
+\begin{tabular}{p{4cm} c c c c}
+Horizontal Grid & \np{jp\_cfg} & \np{jpiglo} & \np{jpjglo} & \\
+\hline \hline
+\~4\deg & 4 & 92 & 76 & \\
+\~2\deg & 2 & 182 & 149 & \\
+\~1\deg & 1 & 362 & 292 & \\
+\~0.5\deg & 05 & 722 & 511 & \\
+\~0.25\deg & 025 & 1442 & 1021 & \\
+%\key{orca\_r8} & 8 & 2882 & 2042 & \\
+%\key{orca\_r12} & 12 & 4322 & 3062 & \\
+\hline \hline
+\end{tabular}
+\caption{ \label{Tab_ORCA}
+Set of predefined parameters for ORCA family configurations.
+In all cases, the name of the configuration is set to "orca" ($i.e.$ \np{cp\_cfg}~=~orca). }
+\end{center}
+\end{table}
+%--------------------------------------------------------------------------------------------------------------
+
+
+The ORCA\_R2 configuration has the following specificity : starting from a 2\deg~ORCA mesh,
+local mesh refinements were applied to the Mediterranean, Red, Black and Caspian Seas,
+so that the resolution is $1\deg \time 1\deg$ there. A local transformation were also applied
+with in the Tropics in order to refine the meridional resolution up to 0.5\deg at the Equator.
+
+The ORCA\_R1 configuration has only a local tropical transformation to refine the meridional
+resolution up to 1/3\deg~at the Equator. Note that the tropical mesh refinements in ORCA\_R2
+and R1 strongly increases the mesh anisotropy there.
+
+The ORCA\_R05 and higher global configurations do not incorporate any regional refinements.
+
+For ORCA\_R1 and R025, setting the configuration key to 75 allows to use 75 vertical levels,
+otherwise 46 are used. In the other ORCA configurations, 31 levels are used
+(see Tab.~\ref{Tab_orca_zgr} and Fig.~\ref{Fig_zgr}).
+
+Only the ORCA\_R2 is provided with all its input files in the \NEMO distribution.
+It is very similar to that used as part of the climate model developed at IPSL for the 4th IPCC
+assessment of climate change (Marti et al., 2009). It is also the basis for the \NEMO contribution
+to the Coordinate Ocean-ice Reference Experiments (COREs) documented in \citet{Griffies_al_OM09}.
+
+This version of ORCA\_R2 has 31 levels in the vertical, with the highest resolution (10m)
+in the upper 150m (see Tab.~\ref{Tab_orca_zgr} and Fig.~\ref{Fig_zgr}).
+The bottom topography and the coastlines are derived from the global atlas of Smith and Sandwell (1997).
+The default forcing uses the boundary forcing from \citet{Large_Yeager_Rep04} (see \S\ref{SBC_blk_core}),
+which was developed for the purpose of running global coupled ocean-ice simulations
+without an interactive atmosphere. This \citet{Large_Yeager_Rep04} dataset is available
+through the \href{http://nomads.gfdl.noaa.gov/nomads/forms/mom4/CORE.html}{GFDL web site}.
+The "normal year" of \citet{Large_Yeager_Rep04} has been chosen of the \NEMO distribution
+since release v3.3.
+
+ORCA\_R2 pre-defined configuration can also be run with an AGRIF zoom over the Agulhas
+current area ( \key{agrif} defined) and, by setting the appropriate variables in
+\textit{\&namcfg}, see \textit{CONFIG/SHARED/namelist\_ref}
+a regional Arctic or peri-Antarctic configuration is extracted from an ORCA\_R2 or R05 configurations
+using sponge layers at open boundaries.
+
+% -------------------------------------------------------------------------------------------------------------
+% GYRE family: double gyre basin
+% -------------------------------------------------------------------------------------------------------------
+\section{GYRE family: double gyre basin }
+\label{CFG_gyre}
+
+The GYRE configuration \citep{Levy_al_OM10} has been built to simulate
+the seasonal cycle of a double-gyre box model. It consists in an idealized domain
+similar to that used in the studies of \citet{Drijfhout_JPO94} and \citet{Hazeleger_Drijfhout_JPO98,
+Hazeleger_Drijfhout_JPO99, Hazeleger_Drijfhout_JGR00, Hazeleger_Drijfhout_JPO00},
+over which an analytical seasonal forcing is applied. This allows to investigate the
+spontaneous generation of a large number of interacting, transient mesoscale eddies
+and their contribution to the large scale circulation.
+
+The domain geometry is a closed rectangular basin on the $\beta$-plane centred
+at $\sim 30\deg$N and rotated by 45\deg, 3180~km long, 2120~km wide
+and 4~km deep (Fig.~\ref{Fig_MISC_strait_hand}).
+The domain is bounded by vertical walls and by a flat bottom. The configuration is
+meant to represent an idealized North Atlantic or North Pacific basin.
+The circulation is forced by analytical profiles of wind and buoyancy fluxes.
+The applied forcings vary seasonally in a sinusoidal manner between winter
+and summer extrema \citep{Levy_al_OM10}.
+The wind stress is zonal and its curl changes sign at 22\deg N and 36\deg N.
+It forces a subpolar gyre in the north, a subtropical gyre in the wider part of the domain
+and a small recirculation gyre in the southern corner.
+The net heat flux takes the form of a restoring toward a zonal apparent air
+temperature profile. A portion of the net heat flux which comes from the solar radiation
+is allowed to penetrate within the water column.
+The fresh water flux is also prescribed and varies zonally.
+It is determined such as, at each time step, the basin-integrated flux is zero.
+The basin is initialised at rest with vertical profiles of temperature and salinity
+uniformly applied to the whole domain.
+
+The GYRE configuration is set through the \textit{\&namcfg} namelist defined in the reference
+configuration \textit{CONFIG/GYRE/EXP00/namelist\_cfg}. Its horizontal resolution
+(and thus the size of the domain) is determined by setting \np{jp\_cfg} : \\
+\np{jpiglo} $= 30 \times$ \np{jp\_cfg} + 2 \\
+\np{jpjglo} $= 20 \times$ \np{jp\_cfg} + 2 \\
+Obviously, the namelist parameters have to be adjusted to the chosen resolution, see the Configurations
+pages on the NEMO web site (Using NEMO\/Configurations) .
+In the vertical, GYRE uses the default 30 ocean levels (\pp{jpk}=31) (Fig.~\ref{Fig_zgr}).
+
+The GYRE configuration is also used in benchmark test as it is very simple to increase
+its resolution and as it does not requires any input file. For example, keeping a same model size
+on each processor while increasing the number of processor used is very easy, even though the
+physical integrity of the solution can be compromised.
+
+%>>>>>>>>>>>>>>>>>>>>>>>>>>>>
+\begin{figure}[!t] \begin{center}
+\includegraphics[width=1.0\textwidth]{./TexFiles/Figures/Fig_GYRE.pdf}
+\caption{ \label{Fig_GYRE}
+Snapshot of relative vorticity at the surface of the model domain
+in GYRE R9, R27 and R54. From \citet{Levy_al_OM10}.}
+\end{center} \end{figure}
+%>>>>>>>>>>>>>>>>>>>>>>>>>>>>
+
+% -------------------------------------------------------------------------------------------------------------
+% EEL family configuration
+% -------------------------------------------------------------------------------------------------------------
+\section{EEL family: periodic channel}
+\label{MISC_config_EEL}
+
+\begin{description}
+\item[eel\_r2] to be described....
+\item[eel\_r5]
+\item[eel\_r6]
+\end{description}
+The appropriate \textit{\&namcfg} namelists are available in
+\textit{CONFIG/SHARED/README\_other\_configurations\_namelist\_namcfg}
+% -------------------------------------------------------------------------------------------------------------
+% AMM configuration
+% -------------------------------------------------------------------------------------------------------------
+\section{AMM: atlantic margin configuration }
+\label{MISC_config_AMM}
+
+The AMM, Atlantic Margins Model, is a regional model covering the
+Northwest European Shelf domain on a regular lat-lon grid at
+approximately 12km horizontal resolution. The appropriate
+\textit{\&namcfg} namelist is available in \textit{CONFIG/AMM12/EXP00/namelist\_cfg}.
+It is used to build the correct dimensions of the AMM domain.
+
+This configuration tests several features of NEMO functionality specific
+to the shelf seas.
+In particular, the AMM uses $S$-coordinates in the vertical rather than
+$z$-coordinates and is forced with tidal lateral boundary conditions
+using a flather boundary condition from the BDY module (key\_bdy).
+The AMM configuration uses the GLS (key\_zdfgls) turbulence scheme, the
+VVL non-linear free surface(key\_vvl) and time-splitting
+(key\_dynspg\_ts).
+
+In addition to the tidal boundary condition the model may also take
+open boundary conditions from a North Atlantic model. Boundaries may be
+completely ommited by removing the BDY key (key\_bdy).
+Sample surface fluxes, river forcing and a sample initial restart file
+are included to test a realistic model run. The Baltic boundary is
+included within the river input file and is specified as a river source.
+Unlike ordinary river points the Baltic inputs also include salinity and
+temperature data.
+
Index: /branches/UKMO/dev_r5518_GO6_under_ice_relax_dr_hook/DOC/TexFiles/Chapters/Chap_Conservation.tex
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+
+% ================================================================
+% Invariant of the Equations
+% ================================================================
+\chapter{Invariants of the Primitive Equations}
+\label{Invariant}
+\minitoc
+
+The continuous equations of motion have many analytic properties. Many
+quantities (total mass, energy, enstrophy, etc.) are strictly conserved in
+the inviscid and unforced limit, while ocean physics conserve the total
+quantities on which they act (momentum, temperature, salinity) but dissipate
+their total variance (energy, enstrophy, etc.). Unfortunately, the finite
+difference form of these equations is not guaranteed to retain all these
+important properties. In constructing the finite differencing schemes, we
+wish to ensure that certain integral constraints will be maintained. In
+particular, it is desirable to construct the finite difference equations so
+that horizontal kinetic energy and/or potential enstrophy of horizontally
+non-divergent flow, and variance of temperature and salinity will be
+conserved in the absence of dissipative effects and forcing. \citet{Arakawa1966}
+has first pointed out the advantage of this approach. He showed that if
+integral constraints on energy are maintained, the computation will be free
+of the troublesome "non linear" instability originally pointed out by
+\citet{Phillips1959}. A consistent formulation of the energetic properties is
+also extremely important in carrying out long-term numerical simulations for
+an oceanographic model. Such a formulation avoids systematic errors that
+accumulate with time \citep{Bryan1997}.
+
+The general philosophy of OPA which has led to the discrete formulation
+presented in {\S}II.2 and II.3 is to choose second order non-diffusive
+scheme for advective terms for both dynamical and tracer equations. At this
+level of complexity, the resulting schemes are dispersive schemes.
+Therefore, they require the addition of a diffusive operator to be stable.
+The alternative is to use diffusive schemes such as upstream or flux
+corrected schemes. This last option was rejected because we prefer a
+complete handling of the model diffusion, i.e. of the model physics rather
+than letting the advective scheme produces its own implicit diffusion
+without controlling the space and time structure of this implicit diffusion.
+Note that in some very specific cases as passive tracer studies, the
+positivity of the advective scheme is required. In that case, and in that
+case only, the advective scheme used for passive tracer is a flux correction
+scheme \citep{Marti1992, Levy1996, Levy1998}.
+
+% -------------------------------------------------------------------------------------------------------------
+% Conservation Properties on Ocean Dynamics
+% -------------------------------------------------------------------------------------------------------------
+\section{Conservation Properties on Ocean Dynamics}
+\label{Invariant_dyn}
+
+The non linear term of the momentum equations has been split into a
+vorticity term, a gradient of horizontal kinetic energy and a vertical
+advection term. Three schemes are available for the former (see {\S}~II.2)
+according to the CPP variable defined (default option\textbf{
+}or \textbf{key{\_}vorenergy } or \textbf{key{\_}vorcombined
+} defined). They differ in their conservative
+properties (energy or enstrophy conserving scheme). The two latter terms
+preserve the total kinetic energy: the large scale kinetic energy is also
+preserved in practice. The remaining non-diffusive terms of the momentum
+equation (namely the hydrostatic and surface pressure gradient terms) also
+preserve the total kinetic energy and have no effect on the vorticity of the
+flow.
+
+\textbf{* relative, planetary and total vorticity term:}
+
+Let us define as either the relative, planetary and total potential
+vorticity, i.e. , , and , respectively. The continuous formulation of the
+vorticity term satisfies following integral constraints:
+\begin{equation} \label{Eq_vor_vorticity}
+\int_D {{\textbf {k}}\cdot \frac{1}{e_3 }\nabla \times \left( {\varsigma
+\;{\rm {\bf k}}\times {\textbf {U}}_h } \right)\;dv} =0
+\end{equation}
+
+\begin{equation} \label{Eq_vor_enstrophy}
+if\quad \chi =0\quad \quad \int\limits_D {\varsigma \;{\textbf{k}}\cdot
+\frac{1}{e_3 }\nabla \times \left( {\varsigma {\textbf{k}}\times {\textbf{U}}_h } \right)\;dv} =-\int\limits_D {\frac{1}{2}\varsigma ^2\,\chi \;dv}
+=0
+\end{equation}
+
+\begin{equation} \label{Eq_vor_energy}
+\int_D {{\textbf{U}}_h \times \left( {\varsigma \;{\textbf{k}}\times {\textbf{U}}_h } \right)\;dv} =0
+\end{equation}
+where $dv = e_1\, e_2\, e_3\, di\, dj\, dk$ is the volume element.
+(II.4.1a) means that $\varsigma $ is conserved. (II.4.1b) is obtained by an
+integration by part. It means that $\varsigma^2$ is conserved for a horizontally
+non-divergent flow.
+(II.4.1c) is even satisfied locally since the vorticity term is orthogonal
+to the horizontal velocity. It means that the vorticity term has no
+contribution to the evolution of the total kinetic energy. (II.4.1a) is
+obviously always satisfied, but (II.4.1b) and (II.4.1c) cannot be satisfied
+simultaneously with a second order scheme. Using the symmetry or
+anti-symmetry properties of the operators (Eqs II.1.10 and 11), it can be
+shown that the scheme (II.2.11) satisfies (II.4.1b) but not (II.4.1c), while
+scheme (II.2.12) satisfies (II.4.1c) but not (II.4.1b) (see appendix C).
+Note that the enstrophy conserving scheme on total vorticity has been chosen
+as the standard discrete form of the vorticity term.
+
+\textbf{* Gradient of kinetic energy / vertical advection}
+
+In continuous formulation, the gradient of horizontal kinetic energy has no
+contribution to the evolution of the vorticity as the curl of a gradient is
+zero. This property is satisfied locally with the discrete form of both the
+gradient and the curl operator we have made (property (II.1.9)~). Another
+continuous property is that the change of horizontal kinetic energy due to
+vertical advection is exactly balanced by the change of horizontal kinetic
+energy due to the horizontal gradient of horizontal kinetic energy:
+
+\begin{equation} \label{Eq_keg_zad}
+\int_D {{\textbf{U}}_h \cdot \nabla _h \left( {1/2\;{\textbf{U}}_h ^2} \right)\;dv} =-\int_D {{\textbf{U}}_h \cdot \frac{w}{e_3 }\;\frac{\partial
+{\textbf{U}}_h }{\partial k}\;dv}
+\end{equation}
+
+Using the discrete form given in {\S}II.2-a and the symmetry or
+anti-symmetry properties of the mean and difference operators, \eqref{Eq_keg_zad} is
+demonstrated in the Appendix C. The main point here is that satisfying
+\eqref{Eq_keg_zad} links the choice of the discrete forms of the vertical advection
+and of the horizontal gradient of horizontal kinetic energy. Choosing one
+imposes the other. The discrete form of the vertical advection given in
+{\S}II.2-a is a direct consequence of formulating the horizontal kinetic
+energy as $1/2 \left( \overline{u^2}^i + \overline{v^2}^j \right) $ in the gradient term.
+
+\textbf{* hydrostatic pressure gradient term}
+
+In continuous formulation, a pressure gradient has no contribution to the
+evolution of the vorticity as the curl of a gradient is zero. This
+properties is satisfied locally with the choice of discretization we have
+made (property (II.1.9)~). In addition, when the equation of state is linear
+(i.e. when an advective-diffusive equation for density can be derived from
+those of temperature and salinity) the change of horizontal kinetic energy
+due to the work of pressure forces is balanced by the change of potential
+energy due to buoyancy forces:
+
+\begin{equation} \label{Eq_hpg_pe}
+\int_D {-\frac{1}{\rho _o }\left. {\nabla p^h} \right|_z \cdot {\textbf {U}}_h \;dv} \;=\;\int_D {\nabla .\left( {\rho \,{\textbf{U}}} \right)\;g\;z\;\;dv}
+\end{equation}
+
+Using the discrete form given in {\S}~II.2-a and the symmetry or
+anti-symmetry properties of the mean and difference operators, (II.4.3) is
+demonstrated in the Appendix C. The main point here is that satisfying
+(II.4.3) strongly constraints the discrete expression of the depth of
+$T$-points and of the term added to the pressure gradient in $s-$coordinates: the
+depth of a $T$-point, $z_T$, is defined as the sum the vertical scale
+factors at $w$-points starting from the surface.
+
+\textbf{* surface pressure gradient term}
+
+In continuous formulation, the surface pressure gradient has no contribution
+to the evolution of vorticity. This properties is trivially satisfied
+locally as (II.2.3) (the equation verified by $\psi$ has been
+derived from the discrete formulation of the momentum equations, vertical
+sum and curl. Nevertheless, the $\psi$-equation is solved numerically by an
+iterative solver (see {\S}~III.5), thus the property is only satisfied with
+the accuracy required on the solver. In addition, with the rigid-lid
+approximation, the change of horizontal kinetic energy due to the work of
+surface pressure forces is exactly zero:
+\begin{equation} \label{Eq_spg}
+\int_D {-\frac{1}{\rho _o }\nabla _h } \left( {p_s } \right)\cdot {\textbf{U}}_h \;dv=0
+\end{equation}
+
+(II.4.4) is satisfied in discrete form only if the discrete barotropic
+streamfunction time evolution equation is given by (II.2.3) (see appendix
+C). This shows that (II.2.3) is the only way to compute the streamfunction,
+otherwise there is no guarantee that the surface pressure force work
+vanishes.
+
+% -------------------------------------------------------------------------------------------------------------
+% Conservation Properties on Ocean Thermodynamics
+% -------------------------------------------------------------------------------------------------------------
+\section{Conservation Properties on Ocean Thermodynamics}
+\label{Invariant_tra}
+
+In continuous formulation, the advective terms of the tracer equations
+conserve the tracer content and the quadratic form of the tracer, i.e.
+\begin{equation} \label{Eq_tra_tra2}
+\int_D {\nabla .\left( {T\;{\textbf{U}}} \right)\;dv} =0
+\;\text{and}
+\int_D {T\;\nabla .\left( {T\;{\textbf{U}}} \right)\;dv} =0
+\end{equation}
+
+The numerical scheme used ({\S}II.2-b) (equations in flux form, second order
+centred finite differences) satisfies (II.4.5) (see appendix C). Note that
+in both continuous and discrete formulations, there is generally no strict
+conservation of mass, since the equation of state is non linear with respect
+to $T$ and $S$. In practice, the mass is conserved with a very good accuracy.
+
+% -------------------------------------------------------------------------------------------------------------
+% Conservation Properties on Momentum Physics
+% -------------------------------------------------------------------------------------------------------------
+\subsection{Conservation Properties on Momentum Physics}
+\label{Invariant_dyn_physics}
+
+\textbf{* lateral momentum diffusion term}
+
+The continuous formulation of the horizontal diffusion of momentum satisfies
+the following integral constraints~:
+\begin{equation} \label{Eq_dynldf_dyn}
+\int\limits_D {\frac{1}{e_3 }{\rm {\bf k}}\cdot \nabla \times \left[ {\nabla
+_h \left( {A^{lm}\;\chi } \right)-\nabla _h \times \left( {A^{lm}\;\zeta
+\;{\rm {\bf k}}} \right)} \right]\;dv} =0
+\end{equation}
+
+\begin{equation} \label{Eq_dynldf_div}
+\int\limits_D {\nabla _h \cdot \left[ {\nabla _h \left( {A^{lm}\;\chi }
+\right)-\nabla _h \times \left( {A^{lm}\;\zeta \;{\rm {\bf k}}} \right)}
+\right]\;dv} =0
+\end{equation}
+
+\begin{equation} \label{Eq_dynldf_curl}
+\int_D {{\rm {\bf U}}_h \cdot \left[ {\nabla _h \left( {A^{lm}\;\chi }
+\right)-\nabla _h \times \left( {A^{lm}\;\zeta \;{\rm {\bf k}}} \right)}
+\right]\;dv} \leqslant 0
+\end{equation}
+
+\begin{equation} \label{Eq_dynldf_curl2}
+\mbox{if}\quad A^{lm}=cste\quad \quad \int_D {\zeta \;{\rm {\bf k}}\cdot
+\nabla \times \left[ {\nabla _h \left( {A^{lm}\;\chi } \right)-\nabla _h
+\times \left( {A^{lm}\;\zeta \;{\rm {\bf k}}} \right)} \right]\;dv}
+\leqslant 0
+\end{equation}
+
+\begin{equation} \label{Eq_dynldf_div2}
+\mbox{if}\quad A^{lm}=cste\quad \quad \int_D {\chi \;\nabla _h \cdot \left[
+{\nabla _h \left( {A^{lm}\;\chi } \right)-\nabla _h \times \left(
+{A^{lm}\;\zeta \;{\rm {\bf k}}} \right)} \right]\;dv} \leqslant 0
+\end{equation}
+
+
+(II.4.6a) and (II.4.6b) means that the horizontal diffusion of momentum
+conserve both the potential vorticity and the divergence of the flow, while
+Eqs (II.4.6c) to (II.4.6e) mean that it dissipates the energy, the enstrophy
+and the square of the divergence. The two latter properties are only
+satisfied when the eddy coefficients are horizontally uniform.
+
+Using (II.1.8) and (II.1.9), and the symmetry or anti-symmetry properties of
+the mean and difference operators, it is shown that the discrete form of the
+lateral momentum diffusion given in {\S}II.2-c satisfies all the integral
+constraints (II.4.6) (see appendix C). In particular, when the eddy
+coefficients are horizontally uniform, a complete separation of vorticity
+and horizontal divergence fields is ensured, so that diffusion (dissipation)
+of vorticity (enstrophy) does not generate horizontal divergence (variance
+of the horizontal divergence) and \textit{vice versa}. When the vertical curl of the horizontal
+diffusion of momentum (discrete sense) is taken, the term associated to the
+horizontal gradient of the divergence is zero locally. When the horizontal
+divergence of the horizontal diffusion of momentum (discrete sense) is
+taken, the term associated to the vertical curl of the vorticity is zero
+locally. The resulting term conserves $\chi$ and dissipates
+$\chi^2$ when the
+eddy coefficient is horizontally uniform.
+
+\textbf{* vertical momentum diffusion term}
+
+As for the lateral momentum physics, the continuous form of the vertical
+diffusion of momentum satisfies following integral constraints~:
+
+conservation of momentum, dissipation of horizontal kinetic energy
+
+\begin{equation} \label{Eq_dynzdf_dyn}
+\begin{aligned}
+& \int_D {\frac{1}{e_3 }} \frac{\partial }{\partial k}\left( \frac{A^{vm}}{e_3 }\frac{\partial {\textbf{U}}_h }{\partial k} \right) \;dv = \overrightarrow{\textbf{0}} \\
+& \int_D \textbf{U}_h \cdot \frac{1}{e_3} \frac{\partial}{\partial k} \left( {\frac{A^{vm}}{e_3 }}{\frac{\partial \textbf{U}_h }{\partial k}} \right) \;dv \leq 0 \\
+ \end{aligned}
+ \end{equation}
+conservation of vorticity, dissipation of enstrophy
+\begin{equation} \label{Eq_dynzdf_vor}
+\begin{aligned}
+& \int_D {\frac{1}{e_3 }{\rm {\bf k}}\cdot \nabla \times \left( {\frac{1}{e_3
+}\frac{\partial }{\partial k}\left( {\frac{A^{vm}}{e_3 }\frac{\partial {\rm
+{\bf U}}_h }{\partial k}} \right)} \right)\;dv} =0 \\
+& \int_D {\zeta \,{\rm {\bf k}}\cdot \nabla \times \left( {\frac{1}{e_3
+}\frac{\partial }{\partial k}\left( {\frac{A^{vm}}{e_3 }\frac{\partial {\rm
+{\bf U}}_h }{\partial k}} \right)} \right)\;dv} \leq 0 \\
+\end{aligned}
+\end{equation}
+conservation of horizontal divergence, dissipation of square of the
+horizontal divergence
+\begin{equation} \label{Eq_dynzdf_div}
+\begin{aligned}
+ &\int_D {\nabla \cdot \left( {\frac{1}{e_3 }\frac{\partial }{\partial
+k}\left( {\frac{A^{vm}}{e_3 }\frac{\partial {\rm {\bf U}}_h }{\partial k}}
+\right)} \right)\;dv} =0 \\
+& \int_D {\chi \;\nabla \cdot \left( {\frac{1}{e_3 }\frac{\partial }{\partial
+k}\left( {\frac{A^{vm}}{e_3 }\frac{\partial {\rm {\bf U}}_h }{\partial k}}
+\right)} \right)\;dv} \leq 0 \\
+\end{aligned}
+\end{equation}
+
+In discrete form, all these properties are satisfied in $z$-coordinate (see
+Appendix C). In $s$-coordinates, only first order properties can be
+demonstrated, i.e. the vertical momentum physics conserve momentum,
+potential vorticity, and horizontal divergence.
+
+% -------------------------------------------------------------------------------------------------------------
+% Conservation Properties on Tracer Physics
+% -------------------------------------------------------------------------------------------------------------
+\subsection{Conservation Properties on Tracer Physics}
+\label{Invariant_tra_physics}
+
+The numerical schemes used for tracer subgridscale physics are written in
+such a way that the heat and salt contents are conserved (equations in flux
+form, second order centred finite differences). As a form flux is used to
+compute the temperature and salinity, the quadratic form of these quantities
+(i.e. their variance) globally tends to diminish. As for the advective term,
+there is generally no strict conservation of mass even if, in practice, the
+mass is conserved with a very good accuracy.
+
+\textbf{* lateral physics: }conservation of tracer, dissipation of tracer
+variance, i.e.
+
+\begin{equation} \label{Eq_traldf_t_t2}
+\begin{aligned}
+&\int_D \nabla\, \cdot\, \left( A^{lT} \,\Re \,\nabla \,T \right)\;dv = 0 \\
+&\int_D \,T\, \nabla\, \cdot\, \left( A^{lT} \,\Re \,\nabla \,T \right)\;dv \leq 0 \\
+\end{aligned}
+\end{equation}
+
+\textbf{* vertical physics: }conservation of tracer, dissipation of tracer
+variance, i.e.
+
+\begin{equation} \label{Eq_trazdf_t_t2}
+\begin{aligned}
+& \int_D \frac{1}{e_3 } \frac{\partial }{\partial k}\left( \frac{A^{vT}}{e_3 } \frac{\partial T}{\partial k} \right)\;dv = 0 \\
+& \int_D \,T \frac{1}{e_3 } \frac{\partial }{\partial k}\left( \frac{A^{vT}}{e_3 } \frac{\partial T}{\partial k} \right)\;dv \leq 0 \\
+\end{aligned}
+\end{equation}
+
+Using the symmetry or anti-symmetry properties of the mean and difference
+operators, it is shown that the discrete form of tracer physics given in
+{\S}~II.2-c satisfies all the integral constraints (II.4.8) and (II.4.9)
+except the dissipation of the square of the tracer when non-geopotential
+diffusion is used (see appendix C). A discrete form of the lateral tracer
+physics can be derived which satisfies these last properties. Nevertheless,
+it requires a horizontal averaging of the vertical component of the lateral
+physics that prevents the use of implicit resolution in the vertical. It has
+not been implemented.
+
Index: /branches/UKMO/dev_r5518_GO6_under_ice_relax_dr_hook/DOC/TexFiles/Chapters/Chap_DIA.tex
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+% ================================================================
+% Chapter I/O & Diagnostics
+% ================================================================
+\chapter{Ouput and Diagnostics (IOM, DIA, TRD, FLO)}
+\label{DIA}
+\minitoc
+
+\newpage
+$\ $\newline % force a new ligne
+
+% ================================================================
+% Old Model Output
+% ================================================================
+\section{Old Model Output (default or \key{dimgout})}
+\label{DIA_io_old}
+
+The model outputs are of three types: the restart file, the output listing,
+and the diagnostic output file(s). The restart file is used internally by the code when
+the user wants to start the model with initial conditions defined by a
+previous simulation. It contains all the information that is necessary in
+order for there to be no changes in the model results (even at the computer
+precision) between a run performed with several restarts and the same run
+performed in one step. It should be noted that this requires that the restart file
+contain two consecutive time steps for all the prognostic variables, and
+that it is saved in the same binary format as the one used by the computer
+that is to read it (in particular, 32 bits binary IEEE format must not be used for
+this file).
+
+The output listing and file(s) are predefined but should be checked
+and eventually adapted to the user's needs. The output listing is stored in
+the $ocean.output$ file. The information is printed from within the code on the
+logical unit $numout$. To locate these prints, use the UNIX command
+"\textit{grep -i numout}" in the source code directory.
+
+By default, diagnostic output files are written in NetCDF format but an IEEE binary output format, called DIMG, can be choosen by defining \key{dimgout}.
+
+Since version 3.2, when defining \key{iomput}, an I/O server has been added which provides more flexibility in the choice of the fields to be written as well as how the writing work is distributed over the processors in massively parallel computing. The complete description of the use of this I/O server is presented in the next section.
+
+By default, if neither \key{iomput} nor \key{dimgout} are defined, NEMO produces NetCDF with the old IOIPSL library which has been kept for compatibility and its easy installation. However, the IOIPSL library is quite inefficient on parallel machines and, since version 3.2, many diagnostic options have been added presuming the use of \key{iomput}. The usefulness of the default IOIPSL-based option is expected to reduce with each new release. If \key{iomput} is not defined, output files and content are defined in the \mdl{diawri} module and contain mean (or instantaneous if \key{diainstant} is defined) values over a regular period of nn\_write time-steps (namelist parameter).
+
+%\gmcomment{ % start of gmcomment
+
+% ================================================================
+% Diagnostics
+% ================================================================
+\section{Standard model Output (IOM)}
+\label{DIA_iom}
+
+
+Since version 3.2, iomput is the NEMO output interface of choice. It has been designed to be simple to use, flexible and efficient. The two main purposes of iomput are:
+\begin{enumerate}
+\item The complete and flexible control of the output files through external XML files adapted by the user from standard templates.
+\item To achieve high performance and scalable output through the optional distribution of all diagnostic output related tasks to dedicated processes.
+\end{enumerate}
+The first functionality allows the user to specify, without code changes or recompilation, aspects of the diagnostic output stream, such as:
+\begin{itemize}
+\item The choice of output frequencies that can be different for each file (including real months and years).
+\item The choice of file contents; includes complete flexibility over which data are written in which files (the same data can be written in different files).
+\item The possibility to split output files at a choosen frequency.
+\item The possibility to extract a vertical or an horizontal subdomain.
+\item The choice of the temporal operation to perform, e.g.: average, accumulate, instantaneous, min, max and once.
+\item Control over metadata via a large XML "database" of possible output fields.
+\end{itemize}
+In addition, iomput allows the user to add the output of any new variable (scalar, 2D or 3D) in the code in a very easy way. All details of iomput functionalities are listed in the following subsections. Examples of the XML files that control the outputs can be found in:
+\begin{alltt}
+\begin{verbatim}
+ NEMOGCM/CONFIG/ORCA2_LIM/EXP00/iodef.xml
+ NEMOGCM/CONFIG/SHARED/field_def.xml
+ and
+ NEMOGCM/CONFIG/SHARED/domain_def.xml.
+\end{verbatim}
+\end{alltt}
+
+The second functionality targets output performance when running in parallel (\key{mpp\_mpi}). Iomput provides the possibility to specify N dedicated I/O processes (in addition to the NEMO processes) to collect and write the outputs. With an appropriate choice of N by the user, the bottleneck associated with the writing of the output files can be greatly reduced.
+
+In version 3.6, the iom\_put interface depends on an external code called \href{https://forge.ipsl.jussieu.fr/ioserver/browser/XIOS/branchs/xios-1.0}{XIOS-1.0} (use of revision 618 or higher is required). This new IO server can take advantage of the parallel I/O functionality of NetCDF4 to create a single output file and therefore to bypass the rebuilding phase. Note that writing in parallel into the same NetCDF files requires that your NetCDF4 library is linked to an HDF5 library that has been correctly compiled (i.e. with the configure option $--$enable-parallel). Note that the files created by iomput through XIOS are incompatible with NetCDF3. All post-processsing and visualization tools must therefore be compatible with NetCDF4 and not only NetCDF3.
+
+Even if not using the parallel I/O functionality of NetCDF4, using N dedicated I/O servers, where N is typically much less than the number of NEMO processors, will reduce the number of output files created. This can greatly reduce the post-processing burden usually associated with using large numbers of NEMO processors. Note that for smaller configurations, the rebuilding phase can be avoided, even without a parallel-enabled NetCDF4 library, simply by employing only one dedicated I/O server.
+
+\subsection{XIOS: the IO\_SERVER}
+
+\subsubsection{Attached or detached mode?}
+
+Iomput is based on \href{http://forge.ipsl.jussieu.fr/ioserver/wiki}{XIOS}, the io\_server developed by Yann Meurdesoif from IPSL. The behaviour of the io subsystem is controlled by settings in the external XML files listed above. Key settings in the iodef.xml file are {\tt using\_server} and the {\tt type} tag associated with each defined file. The {\tt using\_server} setting determines whether or not the server will be used in ''attached mode'' (as a library) [{\tt false}] or in ''detached mode'' (as an external executable on N additional, dedicated cpus) [{\tt true}]. The ''attached mode'' is simpler to use but much less efficient for massively parallel applications. The type of each file can be either ''multiple\_file'' or ''one\_file''.
+
+In attached mode and if the type of file is ''multiple\_file'', then each NEMO process will also act as an IO server and produce its own set of output files. Superficially, this emulates the standard behaviour in previous versions, However, the subdomain written out by each process does not correspond to the {\tt jpi x jpj x jpk} domain actually computed by the process (although it may if {\tt jpni=1}). Instead each process will have collected and written out a number of complete longitudinal strips. If the ''one\_file'' option is chosen then all processes will collect their longitudinal strips and write (in parallel) to a single output file.
+
+In detached mode and if the type of file is ''multiple\_file'', then each stand-alone XIOS process will collect data for a range of complete longitudinal strips and write to its own set of output files. If the ''one\_file'' option is chosen then all XIOS processes will collect their longitudinal strips and write (in parallel) to a single output file. Note running in detached mode requires launching a Multiple Process Multiple Data (MPMD) parallel job. The following subsection provides a typical example but the syntax will vary in different MPP environments.
+
+\subsubsection{Number of cpu used by XIOS in detached mode}
+
+The number of cores used by the XIOS is specified when launching the model. The number of cores dedicated to XIOS should be from ~1/10 to ~1/50 of the number or cores dedicated to NEMO. Some manufacturers suggest using O($\sqrt{N}$) dedicated IO processors for N processors but this is a general recommendation and not specific to NEMO. It is difficult to provide precise recommendations because the optimal choice will depend on the particular hardware properties of the target system (parallel filesystem performance, available memory, memory bandwidth etc.) and the volume and frequency of data to be created. Here is an example of 2 cpus for the io\_server and 62 cpu for nemo using mpirun:
+
+\texttt{ mpirun -np 62 ./nemo.exe : -np 2 ./xios\_server.exe }
+
+\subsubsection{Control of XIOS: the XIOS context in iodef.xml}
+
+As well as the {\tt using\_server} flag, other controls on the use of XIOS are set in the XIOS context in iodef.xml. See the XML basics section below for more details on XML syntax and rules.
+
+\begin{tabular}{|p{4cm}|p{6.0cm}|p{2.0cm}|}
+ \hline
+ variable name &
+ description &
+ example \\
+ \hline
+ \hline
+ buffer\_size &
+ buffer size used by XIOS to send data from NEMO to XIOS. Larger is more efficient. Note that needed/used buffer sizes are summarized at the end of the job &
+ 25000000 \\
+ \hline
+ buffer\_server\_factor\_size &
+ ratio between NEMO and XIOS buffer size. Should be 2. &
+ 2 \\
+ \hline
+ info\_level &
+ verbosity level (0 to 100) &
+ 0 \\
+ \hline
+ using\_server &
+ activate attached(false) or detached(true) mode &
+ true \\
+ \hline
+ using\_oasis &
+ XIOS is used with OASIS(true) or not (false) &
+ false \\
+ \hline
+ oasis\_codes\_id &
+ when using oasis, define the identifier of NEMO in the namcouple. Note that the identifier of XIOS is xios.x &
+ oceanx \\
+ \hline
+\end{tabular}
+
+
+\subsection{Practical issues}
+
+\subsubsection{Installation}
+
+As mentioned, XIOS is supported separately and must be downloaded and compiled before it can be used with NEMO. See the installation guide on the \href{http://forge.ipsl.jussieu.fr/ioserver/wiki}{XIOS} wiki for help and guidance. NEMO will need to link to the compiled XIOS library. The
+\href{http://www.nemo-ocean.eu/Using-NEMO/User-Guides/Basics/XIOS-IO-server-installation-and-use}{XIOS with NEMO} guide provides an example illustration of how this can be achieved.
+
+\subsubsection{Add your own outputs}
+
+It is very easy to add your own outputs with iomput. Many standard fields and diagnostics are already prepared (i.e., steps 1 to 3 below have been done) and simply need to be activated by including the required output in a file definition in iodef.xml (step 4). To add new output variables, all 4 of the following steps must be taken.
+\begin{description}
+\item[1.] in NEMO code, add a \\
+\texttt{ CALL iom\_put( 'identifier', array ) } \\
+where you want to output a 2D or 3D array.
+
+\item[2.] If necessary, add \\
+\texttt{ USE iom\ \ \ \ \ \ \ \ \ \ \ \ ! I/O manager library } \\
+to the list of used modules in the upper part of your module.
+
+\item[3.] in the field\_def.xml file, add the definition of your variable using the same identifier you used in the f90 code (see subsequent sections for a details of the XML syntax and rules). For example:
+\vspace{-20pt}
+\begin{alltt} {{\scriptsize
+\begin{verbatim}
+
+
+
+
+ ...
+
+ ...
+
+\end{verbatim}
+}}\end{alltt}
+Note your definition must be added to the field\_group whose reference grid is consistent with the size of the array passed to iomput. The grid\_ref attribute refers to definitions set in iodef.xml which, in turn, reference grids and axes either defined in the code (iom\_set\_domain\_attr and iom\_set\_axis\_attr in iom.F90) or defined in the domain\_def.xml file. E.g.:
+\vspace{-20pt}
+\begin{alltt} {{\scriptsize
+\begin{verbatim}
+
+\end{verbatim}
+}}\end{alltt}
+Note, if your array is computed within the surface module each nn\_fsbc time\_step,
+add the field definition within the field\_group defined with the id ''SBC'': $<$field\_group id=''SBC''...$>$ which has been defined with the correct frequency of operations (iom\_set\_field\_attr in iom.F90)
+
+\item[4.] add your field in one of the output files defined in iodef.xml (again see subsequent sections for syntax and rules) \\
+\vspace{-20pt}
+\begin{alltt} {{\scriptsize
+\begin{verbatim}
+
+ ...
+
+ ...
+
+\end{verbatim}
+}}\end{alltt}
+
+\end{description}
+\subsection{XML fundamentals}
+
+\subsubsection{ XML basic rules}
+
+XML tags begin with the less-than character ("$<$") and end with the greater-than character (''$>$'').
+You use tags to mark the start and end of elements, which are the logical units of information
+in an XML document. In addition to marking the beginning of an element, XML start tags also
+provide a place to specify attributes. An attribute specifies a single property for an element,
+using a name/value pair, for example: $<$a b="x" c="y" b="z"$>$ ... $<$/a$>$.
+See \href{http://www.xmlnews.org/docs/xml-basics.html}{here} for more details.
+
+\subsubsection{Structure of the xml file used in NEMO}
+
+The XML file used in XIOS is structured by 7 families of tags: context, axis, domain, grid, field, file and variable. Each tag family has hierarchy of three flavors (except for context):
+\\
+\begin{tabular}{|p{3.0cm}|p{4.5cm}|p{4.5cm}|}
+ \hline
+ flavor &
+ description &
+ example \\
+ \hline
+ \hline
+ root &
+ declaration of the root element that can contain element groups or elements &
+ {\scriptsize \verb? < file_definition ... >?} \\
+ \hline
+ group &
+ declaration of a group element that can contain element groups or elements &
+ {\scriptsize \verb? < file_group ... >?} \\
+ \hline
+ element &
+ declaration of an element that can contain elements &
+ {\scriptsize \verb? < file ... >?} \\
+ \hline
+\end{tabular}
+\\
+
+Each element may have several attributes. Some attributes are mandatory, other are optional but have a default value and other are are completely optional. Id is a special attribute used to identify an element or a group of elements. It must be unique for a kind of element. It is optional, but no reference to the corresponding element can be done if it is not defined.
+
+The XML file is split into context tags that are used to isolate IO definition from different codes or different parts of a code. No interference is possible between 2 different contexts. Each context has its own calendar and an associated timestep. In NEMO, we used the following contexts (that can be defined in any order):\\
+\\
+\begin{tabular}{|p{3.0cm}|p{4.5cm}|p{4.5cm}|}
+ \hline
+ context &
+ description &
+ example \\
+ \hline
+ \hline
+ context xios &
+ context containing information for XIOS &
+ {\scriptsize \verb? ?}\\
+
+\noindent In NEMO, by default, the field and domain definition is done in 2 separate files:
+{\scriptsize \tt
+\begin{verbatim}
+NEMOGCM/CONFIG/SHARED/field_def.xml
+and
+NEMOGCM/CONFIG/SHARED/domain_def.xml
+\end{verbatim}
+}
+\noindent that are included in the main iodef.xml file through the following commands: \\
+{\scriptsize \verb? ? \\
+\verb? ? }
+
+
+\subsubsection{Use of inheritance}
+
+XML extensively uses the concept of inheritance. XML has a tree based structure with a parent-child oriented relation: all children inherit attributes from parent, but an attribute defined in a child replace the inherited attribute value. Note that the special attribute ''id'' is never inherited. \\
+\\
+example 1: Direct inheritance.
+\vspace{-20pt}
+\begin{alltt} {{\scriptsize
+\begin{verbatim}
+
+
+
+
+\end{verbatim}
+}}\end{alltt}
+
+The field ''sst'' which is part (or a child) of the field\_definition will inherit the value ''average''
+of the attribute ''operation'' from its parent. Note that a child can overwrite
+the attribute definition inherited from its parents. In the example above, the field ''sss'' will
+for example output instantaneous values instead of average values. \\
+\\
+example 2: Inheritance by reference.
+\vspace{-20pt}
+\begin{alltt} {{\scriptsize
+\begin{verbatim}
+
+
+
+
+
+
+
+
+
+
+
+\end{verbatim}
+}}\end{alltt}
+Inherit (and overwrite, if needed) the attributes of a tag you are refering to.
+
+\subsubsection{Use of Groups}
+
+Groups can be used for 2 purposes. Firstly, the group can be used to define common attributes to be shared by the elements of the group through inheritance. In the following example, we define a group of field that will share a common grid ''grid\_T\_2D''. Note that for the field ''toce'', we overwrite the grid definition inherited from the group by ''grid\_T\_3D''.
+\vspace{-20pt}
+\begin{alltt} {{\scriptsize
+\begin{verbatim}
+
+
+
+
+
+ ...
+\end{verbatim}
+}}\end{alltt}
+
+Secondly, the group can be used to replace a list of elements. Several examples of groups of fields are proposed at the end of the file {\tt CONFIG/SHARED/field\_def.xml}. For example, a short list of the usual variables related to the U grid:
+\vspace{-20pt}
+\begin{alltt} {{\scriptsize
+\begin{verbatim}
+
+
+
+
+
+\end{verbatim}
+}}\end{alltt}
+that can be directly included in a file through the following syntax:
+\vspace{-20pt}
+\begin{alltt} {{\scriptsize
+\begin{verbatim}
+
+
+
+
+\end{verbatim}
+}}\end{alltt}
+
+\subsection{Detailed functionalities }
+
+The file {\tt NEMOGCM/CONFIG/ORCA2\_LIM/iodef\_demo.xml} provides several examples of the use of the new functionalities offered by the XML interface of XIOS.
+
+\subsubsection{Define horizontal subdomains}
+Horizontal subdomains are defined through the attributs zoom\_ibegin, zoom\_jbegin, zoom\_ni, zoom\_nj of the tag family domain. It must therefore be done in the domain part of the XML file. For example, in {\tt CONFIG/SHARED/domain\_def.xml}, we provide the following example of a definition of a 5 by 5 box with the bottom left corner at point (10,10).
+\vspace{-20pt}
+\begin{alltt} {{\scriptsize
+\begin{verbatim}
+
+
+\end{verbatim}
+}}\end{alltt}
+The use of this subdomain is done through the redefinition of the attribute domain\_ref of the tag family field. For example:
+\vspace{-20pt}
+\begin{alltt} {{\scriptsize
+\begin{verbatim}
+
+
+
+\end{verbatim}
+}}\end{alltt}
+Moorings are seen as an extrem case corresponding to a 1 by 1 subdomain. The Equatorial section, the TAO, RAMA and PIRATA moorings are alredy registered in the code and can therefore be outputted without taking care of their (i,j) position in the grid. These predefined domains can be activated by the use of specific domain\_ref: ''EqT'', ''EqU'' or ''EqW'' for the equatorial sections and the mooring position for TAO, RAMA and PIRATA followed by ''T'' (for example: ''8s137eT'', ''1.5s80.5eT'' ...)
+\vspace{-20pt}
+\begin{alltt} {{\scriptsize
+\begin{verbatim}
+
+
+
+\end{verbatim}
+}}\end{alltt}
+Note that if the domain decomposition used in XIOS cuts the subdomain in several parts and if you use the ''multiple\_file'' type for your output files, you will endup with several files you will need to rebuild using unprovided tools (like ncpdq and ncrcat, \href{http://nco.sourceforge.net/nco.html#Concatenation}{see nco manual}). We are therefore advising to use the ''one\_file'' type in this case.
+
+\subsubsection{Define vertical zooms}
+Vertical zooms are defined through the attributs zoom\_begin and zoom\_end of the tag family axis. It must therefore be done in the axis part of the XML file. For example, in NEMOGCM/CONFIG/ORCA2\_LIM/iodef\_demo.xml, we provide the following example:
+\vspace{-20pt}
+\begin{alltt} {{\scriptsize
+\begin{verbatim}
+
+
+
+\end{verbatim}
+}}\end{alltt}
+The use of this vertical zoom is done through the redefinition of the attribute axis\_ref of the tag family field. For example:
+\vspace{-20pt}
+\begin{alltt} {{\scriptsize
+\begin{verbatim}
+
+
+
+\end{verbatim}
+}}\end{alltt}
+
+\subsubsection{Control of the output file names}
+
+The output file names are defined by the attributs ''name'' and ''name\_suffix'' of the tag family file. for example:
+\vspace{-20pt}
+\begin{alltt} {{\scriptsize
+\begin{verbatim}
+
+
+ ...
+
+
+ ...
+
+
+\end{verbatim}
+}}\end{alltt}
+However it is often very convienent to define the file name with the name of the experiment, the output file frequency and the date of the beginning and the end of the simulation (which are informations stored either in the namelist or in the XML file). To do so, we added the following rule: if the id of the tag file is ''fileN''(where N = 1 to 999 on 1 to 3 digits) or one of the predefined sections or moorings (see next subsection), the following part of the name and the name\_suffix (that can be inherited) will be automatically replaced by:\\
+\\
+\begin{tabular}{|p{4cm}|p{8cm}|}
+ \hline
+ \centering placeholder string & automatically replaced by \\
+ \hline
+ \hline
+ \centering @expname@ &
+ the experiment name (from cn\_exp in the namelist) \\
+ \hline
+ \centering @freq@ &
+ output frequency (from attribute output\_freq) \\
+ \hline
+ \centering @startdate@ &
+ starting date of the simulation (from nn\_date0 in the restart or the namelist). \verb?yyyymmdd? format \\
+ \hline
+ \centering @startdatefull@ &
+ starting date of the simulation (from nn\_date0 in the restart or the namelist). \verb?yyyymmdd_hh:mm:ss? format \\
+ \hline
+ \centering @enddate@ &
+ ending date of the simulation (from nn\_date0 and nn\_itend in the namelist). \verb?yyyymmdd? format \\
+ \hline
+ \centering @enddatefull@ &
+ ending date of the simulation (from nn\_date0 and nn\_itend in the namelist). \verb?yyyymmdd_hh:mm:ss? format \\
+ \hline
+\end{tabular}\\
+\\
+
+\noindent For example,
+{{\scriptsize
+\begin{verbatim}
+
+\end{verbatim}
+}}
+\noindent with the namelist:
+{{\scriptsize
+\begin{verbatim}
+ cn_exp = "ORCA2"
+ nn_date0 = 19891231
+ ln_rstart = .false.
+\end{verbatim}
+}}
+\noindent will give the following file name radical:
+{{\scriptsize
+\begin{verbatim}
+ myfile_ORCA2_19891231_freq1d
+\end{verbatim}
+}}
+
+\subsubsection{Other controls of the xml attributes from NEMO}
+
+The values of some attributes are defined by subroutine calls within NEMO (calls to iom\_set\_domain\_attr, iom\_set\_axis\_attr and iom\_set\_field\_attr in iom.F90). Any definition given in the xml file will be overwritten. By convention, these attributes are defined to ''auto'' (for string) or ''0000'' (for integer) in the xml file (but this is not necessary).
+
+Here is the list of these attributes:\\
+\\
+\begin{tabular}{|l|c|c|c|}
+ \hline
+ \multicolumn{2}{|c|}{tag ids affected by automatic } & name & attribute value \\
+ \multicolumn{2}{|c|}{definition of some of their attributes } & attribute & \\
+ \hline
+ \hline
+ \multicolumn{2}{|c|}{field\_definition} & freq\_op & \np{rn\_rdt} \\
+ \hline
+ \multicolumn{2}{|c|}{SBC} & freq\_op & \np{rn\_rdt} $\times$ \np{nn\_fsbc} \\
+ \hline
+ \multicolumn{2}{|c|}{ptrc\_T} & freq\_op & \np{rn\_rdt} $\times$ \np{nn\_dttrc} \\
+ \hline
+ \multicolumn{2}{|c|}{diad\_T} & freq\_op & \np{rn\_rdt} $\times$ \np{nn\_dttrc} \\
+ \hline
+ \multicolumn{2}{|c|}{EqT, EqU, EqW} & jbegin, ni, & according to the grid \\
+ \multicolumn{2}{|c|}{ } & name\_suffix & \\
+ \hline
+ \multicolumn{2}{|c|}{TAO, RAMA and PIRATA moorings} & zoom\_ibegin, zoom\_jbegin, & according to the grid \\
+ \multicolumn{2}{|c|}{ } & name\_suffix & \\
+ \hline
+\end{tabular}
+
+\subsubsection{Advanced use of XIOS functionalities}
+
+\subsection{XML reference tables}
+\label{IOM_xmlref}
+
+(1) Simple computation: directly define the computation when refering to the variable in the file definition.
+
+\vspace{-20pt}
+\begin{alltt} {{\scriptsize
+\begin{verbatim}
+ sst + 273.15
+ taum * taum
+ qt - qsr - qns
+\end{verbatim}
+}}\end{alltt}
+
+(2) Simple computation: define a new variable and use it in the file definition.
+
+in field\_definition:
+\vspace{-20pt}
+\begin{alltt} {{\scriptsize
+\begin{verbatim}
+ sst * sst
+\end{verbatim}
+}}\end{alltt}
+in file\_definition:
+\vspace{-20pt}
+\begin{alltt} {{\scriptsize
+\begin{verbatim}
+ sst2
+\end{verbatim}
+}}\end{alltt}
+Note that in this case, the following syntaxe $<$field field\_ref="sst2" /$>$ is not working as sst2 won't be evaluated.
+
+(3) Change of variable precision:
+
+\vspace{-20pt}
+\begin{alltt} {{\scriptsize
+\begin{verbatim}
+
+
+
+
+\end{verbatim}
+}}\end{alltt}
+Note that, then the code is crashing, writting real4 variables forces a numerical convection from real8 to real4 which will create an internal error in NetCDF and will avoid the creation of the output files. Forcing double precision outputs with prec="8" (for example in the field\_definition) will avoid this problem.
+
+(4) add user defined attributes:
+
+\vspace{-20pt}
+\begin{alltt} {{\scriptsize
+\begin{verbatim}
+
+
+
+ blabla
+ 3
+ 5.0
+
+ blabla\_global
+
+
+\end{verbatim}
+}}\end{alltt}
+
+(5) use of the ``@'' function: example 1, weighted temporal average
+
+ - define a new variable in field\_definition
+\vspace{-20pt}
+\begin{alltt} {{\scriptsize
+\begin{verbatim}
+ toce * e3t
+\end{verbatim}
+}}\end{alltt}
+ - use it when defining your file.
+\vspace{-20pt}
+\begin{alltt} {{\scriptsize
+\begin{verbatim}
+
+
+ @toce\_e3t / @e3t
+
+
+\end{verbatim}
+}}\end{alltt}
+The freq\_op="5d" attribute is used to define the operation frequency of the ``@'' function: here 5 day. The temporal operation done by the ``@'' is the one defined in the field definition: here we use the default, average. So, in the above case, @toce\_e3t will do the 5-day mean of toce*e3t. Operation="instant" refers to the temporal operation to be performed on the field''@toce\_e3t / @e3t'': here the temporal average is alreday done by the ``@'' function so we just use instant to do the ratio of the 2 mean values. field\_ref="toce" means that attributes not explicitely defined, are inherited from toce field. Note that in this case, freq\_op must be equal to the file output\_freq.
+
+(6) use of the ``@'' function: example 2, monthly SSH standard deviation
+
+ - define a new variable in field\_definition
+\vspace{-20pt}
+\begin{alltt} {{\scriptsize
+\begin{verbatim}
+ ssh * ssh
+\end{verbatim}
+}}\end{alltt}
+ - use it when defining your file.
+\vspace{-20pt}
+\begin{alltt} {{\scriptsize
+\begin{verbatim}
+
+
+ sqrt( @ssh2 - @ssh * @ssh )
+
+
+\end{verbatim}
+}}\end{alltt}
+The freq\_op="1m" attribute is used to define the operation frequency of the ``@'' function: here 1 month. The temporal operation done by the ``@'' is the one defined in the field definition: here we use the default, average. So, in the above case, @ssh2 will do the monthly mean of ssh*ssh. Operation="instant" refers to the temporal operation to be performed on the field ''sqrt( @ssh2 - @ssh * @ssh )'': here the temporal average is alreday done by the ``@'' function so we just use instant. field\_ref="ssh" means that attributes not explicitely defined, are inherited from ssh field. Note that in this case, freq\_op must be equal to the file output\_freq.
+
+(7) use of the ``@'' function: example 3, monthly average of SST diurnal cycle
+
+ - define 2 new variables in field\_definition
+\vspace{-20pt}
+\begin{alltt} {{\scriptsize
+\begin{verbatim}
+
+
+\end{verbatim}
+}}\end{alltt}
+ - use these 2 new variables when defining your file.
+\vspace{-20pt}
+\begin{alltt} {{\scriptsize
+\begin{verbatim}
+
+
+ @sstmax - @sstmin
+
+
+\end{verbatim}
+}}\end{alltt}
+The freq\_op="1d" attribute is used to define the operation frequency of the ``@'' function: here 1 day. The temporal operation done by the ``@'' is the one defined in the field definition: here maximum for sstmax and minimum for sstmin. So, in the above case, @sstmax will do the daily max and @sstmin the daily min. Operation="average" refers to the temporal operation to be performed on the field ``@sstmax - @sstmin'': here monthly mean (of daily max - daily min of the sst). field\_ref="sst" means that attributes not explicitely defined, are inherited from sst field.
+
+
+
+\subsubsection{Tag list}
+
+\begin{longtable}{|p{2.2cm}|p{2.5cm}|p{3.5cm}|p{2.2cm}|p{1.6cm}|}
+ \hline
+ tag name &
+ description &
+ accepted attribute &
+ child of &
+ parent of \endhead
+ \hline
+ simulation &
+ this tag is the root tag which encapsulates all the content of the xml file &
+ none &
+ none &
+ context \\
+ \hline
+ context &
+ encapsulates parts of the xml file dedicated to different codes or different parts of a code &
+ id (''xios'', ''nemo'' or ''n\_nemo'' for the nth AGRIF zoom), src, time\_origin &
+ simulation &
+ all root tags: ... \_definition \\
+ \hline
+ \hline
+ field\_definition &
+ encapsulates the definition of all the fields that can potentially be outputted &
+ axis\_ref, default\_value, domain\_ref, enabled, grid\_ref, level, operation, prec, src &
+ context &
+ field or field\_group \\
+ \hline
+ field\_group &
+ encapsulates a group of fields &
+ axis\_ref, default\_value, domain\_ref, enabled, group\_ref, grid\_ref, id, level, operation, prec, src &
+ field\_definition, field\_group, file &
+ field or field\_group \\
+ \hline
+ field &
+ define a specific field &
+ axis\_ref, default\_value, domain\_ref, enabled, field\_ref, grid\_ref, id, level, long\_name, name, operation, prec, standard\_name, unit &
+ field\_definition, field\_group, file &
+ none \\
+ \hline
+ \hline
+ file\_definition &
+ encapsulates the definition of all the files that will be outputted &
+ enabled, min\_digits, name, name\_suffix, output\_level, split\_freq\_format, split\_freq, sync\_freq, type, src &
+ context &
+ file or file\_group \\
+ \hline
+ file\_group &
+ encapsulates a group of files that will be outputted &
+ enabled, description, id, min\_digits, name, name\_suffix, output\_freq, output\_level, split\_freq\_format, split\_freq, sync\_freq, type, src &
+ file\_definition, file\_group &
+ file or file\_group \\
+ \hline
+ file &
+ define the contents of a file to be outputted &
+ enabled, description, id, min\_digits, name, name\_suffix, output\_freq, output\_level, split\_freq\_format, split\_freq, sync\_freq, type, src &
+ file\_definition, file\_group &
+ field \\
+ \hline
+ axis\_definition &
+ define all the vertical axis potentially used by the variables &
+ src &
+ context &
+ axis\_group, axis \\
+ \hline
+ axis\_group &
+ encapsulates a group of vertical axis &
+ id, lon\_name, positive, src, standard\_name, unit, zoom\_begin, zoom\_end, zoom\_size &
+ axis\_definition, axis\_group &
+ axis\_group, axis \\
+ \hline
+ axis &
+ define a vertical axis &
+ id, lon\_name, positive, src, standard\_name, unit, zoom\_begin, zoom\_end, zoom\_size &
+ axis\_definition, axis\_group &
+ none \\
+ \hline
+ \hline
+ domain\_\-definition &
+ define all the horizontal domains potentially used by the variables &
+ src &
+ context &
+ domain\_\-group, domain \\
+ \hline
+ domain\_group &
+ encapsulates a group of horizontal domains &
+ id, lon\_name, src, zoom\_ibegin, zoom\_jbegin, zoom\_ni, zoom\_nj &
+ domain\_\-definition, domain\_group &
+ domain\_\-group, domain \\
+ \hline
+ domain &
+ define an horizontal domain &
+ id, lon\_name, src, zoom\_ibegin, zoom\_jbegin, zoom\_ni, zoom\_nj &
+ domain\_\-definition, domain\_group &
+ none \\
+ \hline
+ \hline
+ grid\_definition &
+ define all the grid (association of a domain and/or an axis) potentially used by the variables &
+ src &
+ context &
+ grid\_group, grid \\
+ \hline
+ grid\_group &
+ encapsulates a group of grids &
+ id, domain\_ref, axis\_ref &
+ grid\_definition, grid\_group &
+ grid\_group, grid \\
+ \hline
+ grid &
+ define a grid &
+ id, domain\_ref, axis\_ref &
+ grid\_definition, grid\_group &
+ none \\
+ \hline
+\end{longtable}
+
+
+\subsubsection{Attributes list}
+
+\begin{longtable}{|p{2.2cm}|p{4cm}|p{3.8cm}|p{2cm}|}
+ \hline
+ attribute name &
+ description &
+ example &
+ accepted by \endhead
+ \hline
+ axis\_ref &
+ refers to the id of a vertical axis &
+ axis\_ref="deptht" &
+ field, grid families \\
+ \hline
+ enabled &
+ switch on/off the output of a field or a file &
+ enabled=".TRUE." &
+ field, file families \\
+ \hline
+ default\_value &
+ missing\_value definition &
+ default\_value="1.e20" &
+ field family \\
+ \hline
+ description &
+ just for information, not used &
+ description="ocean T grid variables" &
+ all tags \\
+ \hline
+ domain\_ref &
+ refers to the id of a domain &
+ domain\_ref="grid\_T" &
+ field or grid families \\
+ \hline
+ field\_ref &
+ id of the field we want to add in a file &
+ field\_ref="toce" &
+ field \\
+ \hline
+ grid\_ref &
+ refers to the id of a grid &
+ grid\_ref="grid\_T\_2D" &
+ field family \\
+ \hline
+ group\_ref &
+ refer to a group of variables &
+ group\_ref="mooring" &
+ field\_group \\
+ \hline
+ id &
+ allow to identify a tag &
+ id="nemo" &
+ accepted by all tags except simulation \\
+ \hline
+ level &
+ output priority of a field: 0 (high) to 10 (low)&
+ level="1" &
+ field family \\
+ \hline
+ long\_name &
+ define the long\_name attribute in the NetCDF file &
+ long\_name="Vertical T levels" &
+ field \\
+ \hline
+ min\_digits &
+ specify the minimum of digits used in the core number in the name of the NetCDF file &
+ min\_digits="4" &
+ file family \\
+ \hline
+ name &
+ name of a variable or a file. If the name of a file is undefined, its id is used as a name &
+ name="tos" &
+ field or file families \\
+ \hline
+ name\_suffix &
+ suffix to be inserted after the name and before the cpu number and the ''.nc'' termination of a file &
+ name\_suffix="\_myzoom" &
+ file family \\
+ \hline
+ attribute name &
+ description &
+ example &
+ accepted by \\
+ \hline
+ \hline
+ operation &
+ type of temporal operation: average, accumulate, instantaneous, min, max and once &
+ operation="average" &
+ field family \\
+ \hline
+ output\_freq &
+ operation frequency. units can be ts (timestep), y, mo, d, h, mi, s. &
+ output\_freq="1d12h" &
+ field family \\
+ \hline
+ output\_level &
+ output priority of variables in a file: 0 (high) to 10 (low). All variables listed in the file with a level smaller or equal to output\_level will be output. Other variables won't be output even if they are listed in the file. &
+ output\_level="10"&
+ file family \\
+ \hline
+ positive &
+ convention used for the orientation of vertival axis (positive downward in \NEMO). &
+ positive="down" &
+ axis family \\
+ \hline
+ prec &
+ output precision: real 4 or real 8 &
+ prec="4" &
+ field family \\
+ \hline
+ split\_freq &
+ frequency at which to temporally split output files. Units can be ts (timestep), y, mo, d, h, mi, s. Useful for long runs to prevent over-sized output files.&
+ split\_freq="1mo" &
+ file family \\
+ \hline
+ split\_freq\-\_format &
+ date format used in the name of temporally split output files. Can be specified
+ using the following syntaxes: \%y, \%mo, \%d, \%h \%mi and \%s &
+ split\_freq\_format= "\%y\%mo\%d" &
+ file family \\
+ \hline
+ src &
+ allow to include a file &
+ src="./field\_def.xml" &
+ accepted by all tags except simulation \\
+ \hline
+ standard\_name &
+ define the standard\_name attribute in the NetCDF file &
+ standard\_name= "Eastward\_Sea\_Ice\_Transport" &
+ field \\
+ \hline
+ sync\_freq &
+ NetCDF file synchronization frequency (update of the time\_counter). units can be ts (timestep), y, mo, d, h, mi, s. &
+ sync\_freq="10d" &
+ file family \\
+ \hline
+ attribute name &
+ description &
+ example &
+ accepted by \\
+ \hline
+ \hline
+ time\_origin &
+ specify the origin of the time counter &
+ time\_origin="1900-01-01 00:00:00"&
+ context \\
+ \hline
+ type (1)&
+ specify if the output files are to be split spatially (multiple\_file) or not (one\_file) &
+ type="multiple\_file" &
+ file familly \\
+ \hline
+ type (2)&
+ define the type of a variable tag &
+ type="boolean" &
+ variable \\
+ \hline
+ unit &
+ unit of a variable or the vertical axis &
+ unit="m" &
+ field and axis families \\
+ \hline
+ zoom\_ibegin &
+ starting point along x direction of the zoom. Automatically defined for TAO/RAMA/PIRATA moorings &
+ zoom\_ibegin="1" &
+ domain family \\
+ \hline
+ zoom\_jbegin &
+ starting point along y direction of the zoom. Automatically defined for TAO/RAMA/PIRATA moorings &
+ zoom\_jbegin="1" &
+ domain family \\
+ \hline
+ zoom\_ni &
+ zoom extent along x direction &
+ zoom\_ni="1" &
+ domain family \\
+ \hline
+ zoom\_nj &
+ zoom extent along y direction &
+ zoom\_nj="1" &
+ domain family \\
+ \hline
+\end{longtable}
+
+\subsection{CF metadata standard compliance}
+
+Output from the XIOS-1.0 IO server is compliant with \href{http://cfconventions.org/Data/cf-conventions/cf-conventions-1.5/build/cf-conventions.html}{version 1.5} of the CF metadata standard. Therefore while a user may wish to add their own metadata to the output files (as demonstrated in example 4 of section \ref{IOM_xmlref}) the metadata should, for the most part, comply with the CF-1.5 standard.
+
+Some metadata that may significantly increase the file size (horizontal cell areas and vertices) are controlled by the namelist parameter \np{ln\_cfmeta} in the \ngn{namrun} namelist. This must be set to true if these metadata are to be included in the output files.
+
+
+% ================================================================
+% NetCDF4 support
+% ================================================================
+\section{NetCDF4 Support (\key{netcdf4})}
+\label{DIA_iom}
+
+Since version 3.3, support for NetCDF4 chunking and (loss-less) compression has
+been included. These options build on the standard NetCDF output and allow
+the user control over the size of the chunks via namelist settings. Chunking
+and compression can lead to significant reductions in file sizes for a small
+runtime overhead. For a fuller discussion on chunking and other performance
+issues the reader is referred to the NetCDF4 documentation found
+\href{http://www.unidata.ucar.edu/software/netcdf/docs/netcdf.html#Chunking}{here}.
+
+The new features are only available when the code has been linked with a
+NetCDF4 library (version 4.1 onwards, recommended) which has been built
+with HDF5 support (version 1.8.4 onwards, recommended). Datasets created
+with chunking and compression are not backwards compatible with NetCDF3
+"classic" format but most analysis codes can be relinked simply with the
+new libraries and will then read both NetCDF3 and NetCDF4 files. NEMO
+executables linked with NetCDF4 libraries can be made to produce NetCDF3
+files by setting the \np{ln\_nc4zip} logical to false in the \textit{namnc4}
+namelist:
+
+%------------------------------------------namnc4----------------------------------------------------
+\namdisplay{namnc4}
+%-------------------------------------------------------------------------------------------------------------
+
+If \key{netcdf4} has not been defined, these namelist parameters are not read.
+In this case, \np{ln\_nc4zip} is set false and dummy routines for a few
+NetCDF4-specific functions are defined. These functions will not be used but
+need to be included so that compilation is possible with NetCDF3 libraries.
+
+When using NetCDF4 libraries, \key{netcdf4} should be defined even if the
+intention is to create only NetCDF3-compatible files. This is necessary to
+avoid duplication between the dummy routines and the actual routines present
+in the library. Most compilers will fail at compile time when faced with
+such duplication. Thus when linking with NetCDF4 libraries the user must
+define \key{netcdf4} and control the type of NetCDF file produced via the
+namelist parameter.
+
+Chunking and compression is applied only to 4D fields and there is no
+advantage in chunking across more than one time dimension since previously
+written chunks would have to be read back and decompressed before being
+added to. Therefore, user control over chunk sizes is provided only for the
+three space dimensions. The user sets an approximate number of chunks along
+each spatial axis. The actual size of the chunks will depend on global domain
+size for mono-processors or, more likely, the local processor domain size for
+distributed processing. The derived values are subject to practical minimum
+values (to avoid wastefully small chunk sizes) and cannot be greater than the
+domain size in any dimension. The algorithm used is:
+
+\vspace{-20pt}
+\begin{alltt} {{\scriptsize
+\begin{verbatim}
+ ichunksz(1) = MIN( idomain_size,MAX( (idomain_size-1)/nn_nchunks_i + 1 ,16 ) )
+ ichunksz(2) = MIN( jdomain_size,MAX( (jdomain_size-1)/nn_nchunks_j + 1 ,16 ) )
+ ichunksz(3) = MIN( kdomain_size,MAX( (kdomain_size-1)/nn_nchunks_k + 1 , 1 ) )
+ ichunksz(4) = 1
+\end{verbatim}
+}}\end{alltt}
+
+\noindent As an example, setting:
+\vspace{-20pt}
+\begin{alltt} {{\scriptsize
+\begin{verbatim}
+ nn_nchunks_i=4, nn_nchunks_j=4 and nn_nchunks_k=31
+\end{verbatim}
+}}\end{alltt} \vspace{-10pt}
+
+\noindent for a standard ORCA2\_LIM configuration gives chunksizes of {\small\tt 46x38x1}
+respectively in the mono-processor case (i.e. global domain of {\small\tt 182x149x31}).
+An illustration of the potential space savings that NetCDF4 chunking and compression
+provides is given in table \ref{Tab_NC4} which compares the results of two short
+runs of the ORCA2\_LIM reference configuration with a 4x2 mpi partitioning. Note
+the variation in the compression ratio achieved which reflects chiefly the dry to wet
+volume ratio of each processing region.
+
+%------------------------------------------TABLE----------------------------------------------------
+\begin{table} \begin{tabular}{lrrr}
+Filename & NetCDF3 & NetCDF4 & Reduction\\
+ &filesize & filesize & \% \\
+ &(KB) & (KB) & \\
+ORCA2\_restart\_0000.nc & 16420 & 8860 & 47\%\\
+ORCA2\_restart\_0001.nc & 16064 & 11456 & 29\%\\
+ORCA2\_restart\_0002.nc & 16064 & 9744 & 40\%\\
+ORCA2\_restart\_0003.nc & 16420 & 9404 & 43\%\\
+ORCA2\_restart\_0004.nc & 16200 & 5844 & 64\%\\
+ORCA2\_restart\_0005.nc & 15848 & 8172 & 49\%\\
+ORCA2\_restart\_0006.nc & 15848 & 8012 & 50\%\\
+ORCA2\_restart\_0007.nc & 16200 & 5148 & 69\%\\
+ORCA2\_2d\_grid\_T\_0000.nc & 2200 & 1504 & 32\%\\
+ORCA2\_2d\_grid\_T\_0001.nc & 2200 & 1748 & 21\%\\
+ORCA2\_2d\_grid\_T\_0002.nc & 2200 & 1592 & 28\%\\
+ORCA2\_2d\_grid\_T\_0003.nc & 2200 & 1540 & 30\%\\
+ORCA2\_2d\_grid\_T\_0004.nc & 2200 & 1204 & 46\%\\
+ORCA2\_2d\_grid\_T\_0005.nc & 2200 & 1444 & 35\%\\
+ORCA2\_2d\_grid\_T\_0006.nc & 2200 & 1428 & 36\%\\
+ORCA2\_2d\_grid\_T\_0007.nc & 2200 & 1148 & 48\%\\
+ ... & ... & ... & .. \\
+ORCA2\_2d\_grid\_W\_0000.nc & 4416 & 2240 & 50\%\\
+ORCA2\_2d\_grid\_W\_0001.nc & 4416 & 2924 & 34\%\\
+ORCA2\_2d\_grid\_W\_0002.nc & 4416 & 2512 & 44\%\\
+ORCA2\_2d\_grid\_W\_0003.nc & 4416 & 2368 & 47\%\\
+ORCA2\_2d\_grid\_W\_0004.nc & 4416 & 1432 & 68\%\\
+ORCA2\_2d\_grid\_W\_0005.nc & 4416 & 1972 & 56\%\\
+ORCA2\_2d\_grid\_W\_0006.nc & 4416 & 2028 & 55\%\\
+ORCA2\_2d\_grid\_W\_0007.nc & 4416 & 1368 & 70\%\\
+\end{tabular}
+\caption{ \label{Tab_NC4}
+Filesize comparison between NetCDF3 and NetCDF4 with chunking and compression}
+\end{table}
+%----------------------------------------------------------------------------------------------------
+
+When \key{iomput} is activated with \key{netcdf4} chunking and
+compression parameters for fields produced via \np{iom\_put} calls are
+set via an equivalent and identically named namelist to \textit{namnc4}
+in \np{xmlio\_server.def}. Typically this namelist serves the mean files
+whilst the \ngn{ namnc4} in the main namelist file continues to serve the
+restart files. This duplication is unfortunate but appropriate since, if
+using io\_servers, the domain sizes of the individual files produced by the
+io\_server processes may be different to those produced by the invidual
+processing regions and different chunking choices may be desired.
+
+
+% -------------------------------------------------------------------------------------------------------------
+% Tracer/Dynamics Trends
+% -------------------------------------------------------------------------------------------------------------
+\section[Tracer/Dynamics Trends (TRD)]
+ {Tracer/Dynamics Trends (\key{trdtra}, \key{trddyn}, \\
+ \key{trddvor}, \key{trdmld})}
+\label{DIA_trd}
+
+%------------------------------------------namtrd----------------------------------------------------
+\namdisplay{namtrd}
+%-------------------------------------------------------------------------------------------------------------
+
+When \key{trddyn} and/or \key{trddyn} CPP variables are defined, each
+trend of the dynamics and/or temperature and salinity time evolution equations
+is stored in three-dimensional arrays just after their computation ($i.e.$ at the end
+of each $dyn\cdots.F90$ and/or $tra\cdots.F90$ routines). Options are defined by
+\ngn{namtrd} namelist variables. These trends are then
+used in \mdl{trdmod} (see TRD directory) every \textit{nn\_trd } time-steps.
+
+What is done depends on the CPP keys defined:
+\begin{description}
+\item[\key{trddyn}, \key{trdtra}] : a check of the basin averaged properties of the momentum
+and/or tracer equations is performed ;
+\item[\key{trdvor}] : a vertical summation of the moment tendencies is performed,
+then the curl is computed to obtain the barotropic vorticity tendencies which are output ;
+\item[\key{trdmld}] : output of the tracer tendencies averaged vertically
+either over the mixed layer (\np{nn\_ctls}=0),
+or over a fixed number of model levels (\np{nn\_ctls}$>$1 provides the number of level),
+or over a spatially varying but temporally fixed number of levels (typically the base
+of the winter mixed layer) read in \ifile{ctlsurf\_idx} (\np{nn\_ctls}=1) ;
+\end{description}
+
+The units in the output file can be changed using the \np{nn\_ucf} namelist parameter.
+For example, in case of salinity tendency the units are given by PSU/s/\np{nn\_ucf}.
+Setting \np{nn\_ucf}=86400 ($i.e.$ the number of second in a day) provides the tendencies in PSU/d.
+
+When \key{trdmld} is defined, two time averaging procedure are proposed.
+Setting \np{ln\_trdmld\_instant} to \textit{true}, a simple time averaging is performed,
+so that the resulting tendency is the contribution to the change of a quantity between
+the two instantaneous values taken at the extremities of the time averaging period.
+Setting \np{ln\_trdmld\_instant} to \textit{false}, a double time averaging is performed,
+so that the resulting tendency is the contribution to the change of a quantity between
+two \textit{time mean} values. The later option requires the use of an extra file, \ifile{restart\_mld}
+(\np{ln\_trdmld\_restart}=true), to restart a run.
+
+
+Note that the mixed layer tendency diagnostic can also be used on biogeochemical models
+via the \key{trdtrc} and \key{trdmld\_trc} CPP keys.
+
+% -------------------------------------------------------------------------------------------------------------
+% On-line Floats trajectories
+% -------------------------------------------------------------------------------------------------------------
+\section{On-line Floats trajectories (FLO) (\key{floats})}
+\label{FLO}
+%--------------------------------------------namflo-------------------------------------------------------
+\namdisplay{namflo}
+%--------------------------------------------------------------------------------------------------------------
+
+The on-line computation of floats advected either by the three dimensional velocity
+field or constraint to remain at a given depth ($w = 0$ in the computation) have been
+introduced in the system during the CLIPPER project. Options are defined by \ngn{namflo}
+namelis variables. The algorithm used is based
+either on the work of \cite{Blanke_Raynaud_JPO97} (default option), or on a $4^th$
+Runge-Hutta algorithm (\np{ln\_flork4}=true). Note that the \cite{Blanke_Raynaud_JPO97}
+algorithm have the advantage of providing trajectories which are consistent with the
+numeric of the code, so that the trajectories never intercept the bathymetry.
+
+\subsubsection{ Input data: initial coordinates }
+
+Initial coordinates can be given with Ariane Tools convention ( IJK coordinates ,(\np{ln\_ariane}=true) )
+or with longitude and latitude.
+
+
+In case of Ariane convention, input filename is \np{init\_float\_ariane}. Its format is:
+
+\texttt{ I J K nisobfl itrash itrash }
+
+\noindent with:
+
+ - I,J,K : indexes of initial position
+
+ - nisobfl: 0 for an isobar float, 1 for a float following the w velocity
+
+ - itrash : set to zero; it is a dummy variable to respect Ariane Tools convention
+
+ - itrash :set to zero; it is a dummy variable to respect Ariane Tools convention
+
+\noindent Example:\\
+\noindent \texttt{ 100.00000 90.00000 -1.50000 1.00000 0.00000}\\
+\texttt{ 102.00000 90.00000 -1.50000 1.00000 0.00000}\\
+\texttt{ 104.00000 90.00000 -1.50000 1.00000 0.00000}\\
+\texttt{ 106.00000 90.00000 -1.50000 1.00000 0.00000}\\
+\texttt{ 108.00000 90.00000 -1.50000 1.00000 0.00000}\\
+
+
+In the other case ( longitude and latitude ), input filename is init\_float. Its format is:
+
+\texttt{ Long Lat depth nisobfl ngrpfl itrash}
+
+\noindent with:
+
+ - Long, Lat, depth : Longitude, latitude, depth
+
+ - nisobfl: 0 for an isobar float, 1 for a float following the w velocity
+
+ - ngrpfl : number to identify searcher group
+
+ - itrash :set to 1; it is a dummy variable.
+
+\noindent Example:
+
+\noindent\texttt{ 20.0 0.0 0.0 0 1 1 }\\
+\texttt{ -21.0 0.0 0.0 0 1 1 }\\
+\texttt{ -22.0 0.0 0.0 0 1 1 }\\
+\texttt{ -23.0 0.0 0.0 0 1 1 }\\
+\texttt{ -24.0 0.0 0.0 0 1 1 }\\
+
+\np{jpnfl} is the total number of floats during the run.
+When initial positions are read in a restart file ( \np{ln\_rstflo}= .TRUE. ), \np{jpnflnewflo}
+can be added in the initialization file.
+
+\subsubsection{ Output data }
+
+\np{nn\_writefl} is the frequency of writing in float output file and \np{nn\_stockfl}
+is the frequency of creation of the float restart file.
+
+Output data can be written in ascii files (\np{ln\_flo\_ascii} = .TRUE. ). In that case,
+output filename is trajec\_float.
+
+Another possiblity of writing format is Netcdf (\np{ln\_flo\_ascii} = .FALSE. ). There are 2 possibilities:
+
+ - if (\key{iomput}) is used, outputs are selected in iodef.xml. Here it is an example of specification
+ to put in files description section:
+
+\vspace{-30pt}
+\begin{alltt} {{\scriptsize
+\begin{verbatim}
+
+ }
+ }\\
+ }
+ }
+ }
+ }
+ }
+ }
+ }
+ }
+ }
+
+\end{verbatim}
+}}\end{alltt}
+
+
+ - if (\key{iomput}) is not used, a file called trajec\_float.nc will be created by IOIPSL library.
+
+
+
+See also \href{http://stockage.univ-brest.fr/~grima/Ariane/}{here} the web site describing
+the off-line use of this marvellous diagnostic tool.
+
+
+% -------------------------------------------------------------------------------------------------------------
+% Harmonic analysis of tidal constituents
+% -------------------------------------------------------------------------------------------------------------
+\section{Harmonic analysis of tidal constituents (\key{diaharm}) }
+\label{DIA_diag_harm}
+
+A module is available to compute the amplitude and phase for tidal waves.
+This diagnostic is actived with \key{diaharm}.
+
+%------------------------------------------namdia_harm----------------------------------------------------
+\namdisplay{namdia_harm}
+%----------------------------------------------------------------------------------------------------------
+
+Concerning the on-line Harmonic analysis, some parameters are available in namelist
+\ngn{namdia\_harm} :
+
+- \texttt{nit000\_han} is the first time step used for harmonic analysis
+
+- \texttt{nitend\_han} is the last time step used for harmonic analysis
+
+- \texttt{nstep\_han} is the time step frequency for harmonic analysis
+
+- \texttt{nb\_ana} is the number of harmonics to analyse
+
+- \texttt{tname} is an array with names of tidal constituents to analyse
+
+\texttt{nit000\_han} and \texttt{nitend\_han} must be between \texttt{nit000} and \texttt{nitend} of the simulation.
+The restart capability is not implemented.
+
+The Harmonic analysis solve this equation:
+\begin{equation}
+h_{i} - A_{0} + \sum^{nb\_ana}_{j=1}[A_{j}cos(\nu_{j}t_{j}-\phi_{j})] = e_{i}
+\end{equation}
+
+With $A_{j}$,$\nu_{j}$,$\phi_{j}$, the amplitude, frequency and phase for each wave and $e_{i}$ the error.
+$h_{i}$ is the sea level for the time $t_{i}$ and $A_{0}$ is the mean sea level. \\
+We can rewrite this equation:
+\begin{equation}
+h_{i} - A_{0} + \sum^{nb\_ana}_{j=1}[C_{j}cos(\nu_{j}t_{j})+S_{j}sin(\nu_{j}t_{j})] = e_{i}
+\end{equation}
+with $A_{j}=\sqrt{C^{2}_{j}+S^{2}_{j}}$ et $\phi_{j}=arctan(S_{j}/C_{j})$.
+
+We obtain in output $C_{j}$ and $S_{j}$ for each tidal wave.
+
+% -------------------------------------------------------------------------------------------------------------
+% Sections transports
+% -------------------------------------------------------------------------------------------------------------
+\section{Transports across sections (\key{diadct}) }
+\label{DIA_diag_dct}
+
+A module is available to compute the transport of volume, heat and salt through sections. This diagnostic
+is actived with \key{diadct}.
+
+Each section is defined by the coordinates of its 2 extremities. The pathways between them are contructed
+using tools which can be found in \texttt{NEMOGCM/TOOLS/SECTIONS\_DIADCT} and are written in a binary file
+ \texttt{section\_ijglobal.diadct\_ORCA2\_LIM} which is later read in by NEMO to compute on-line transports.
+
+The on-line transports module creates three output ascii files:
+
+- \texttt{volume\_transport} for volume transports ( unit: $10^{6} m^{3} s^{-1}$ )
+
+- \texttt{heat\_transport} for heat transports ( unit: $10^{15} W $ )
+
+- \texttt{salt\_transport} for salt transports ( unit: $10^{9}Kg s^{-1}$ )\\
+
+
+Namelist variables in \ngn{namdct} control how frequently the flows are summed
+and the time scales over which they are averaged, as well as the level of output for debugging:
+
+%------------------------------------------namdct----------------------------------------------------
+\namdisplay{namdct}
+%-------------------------------------------------------------------------------------------------------------
+
+\texttt{nn\_dct}: frequency of instantaneous transports computing
+
+\texttt{nn\_dctwri}: frequency of writing ( mean of instantaneous transports )
+
+\texttt{nn\_debug}: debugging of the section
+
+\subsubsection{ To create a binary file containing the pathway of each section }
+
+In \texttt{NEMOGCM/TOOLS/SECTIONS\_DIADCT/run}, the file \texttt{ {list\_sections.ascii\_global}}
+contains a list of all the sections that are to be computed (this list of sections is based on MERSEA project metrics).
+
+Another file is available for the GYRE configuration (\texttt{ {list\_sections.ascii\_GYRE}}).
+
+Each section is defined by:
+
+\noindent \texttt{ long1 lat1 long2 lat2 nclass (ok/no)strpond (no)ice section\_name }\\
+with:
+
+- \texttt{long1 lat1} , coordinates of the first extremity of the section;
+
+- \texttt{long2 lat2} , coordinates of the second extremity of the section;
+
+- \texttt{nclass} the number of bounds of your classes (e.g. 3 bounds for 2 classes);
+
+- \texttt{okstrpond} to compute heat and salt transport, \texttt{nostrpond} if no;
+
+- \texttt{ice} to compute surface and volume ice transports, \texttt{noice} if no. \\
+
+
+\noindent The results of the computing of transports, and the directions of positive
+ and negative flow do not depend on the order of the 2 extremities in this file.\\
+
+
+\noindent If nclass =/ 0,the next lines contain the class type and the nclass bounds:
+
+\texttt{long1 lat1 long2 lat2 nclass (ok/no)strpond (no)ice section\_name}
+
+\texttt{classtype}
+
+\texttt{zbound1}
+
+\texttt{zbound2}
+
+\texttt{.}
+
+\texttt{.}
+
+\texttt{nclass-1}
+
+\texttt{nclass}
+
+\noindent where \texttt{classtype} can be:
+
+- \texttt{zsal} for salinity classes
+
+- \texttt{ztem} for temperature classes
+
+- \texttt{zlay} for depth classes
+
+- \texttt{zsigi} for insitu density classes
+
+- \texttt{zsigp} for potential density classes \\
+
+
+The script \texttt{job.ksh} computes the pathway for each section and creates a binary file
+\texttt{section\_ijglobal.diadct\_ORCA2\_LIM} which is read by NEMO. \\
+
+It is possible to use this tools for new configuations: \texttt{job.ksh} has to be updated
+with the coordinates file name and path. \\
+
+
+Examples of two sections, the ACC\_Drake\_Passage with no classes, and the
+ ATL\_Cuba\_Florida with 4 temperature clases (5 class bounds), are shown:
+
+\noindent \texttt{ -68. -54.5 -60. -64.7 00 okstrpond noice ACC\_Drake\_Passage}
+
+\noindent \texttt{ -80.5 22.5 -80.5 25.5 05 nostrpond noice ATL\_Cuba\_Florida}
+
+\noindent \texttt{ztem}
+
+\noindent \texttt{-2.0}
+
+\noindent \texttt{ 4.5}
+
+\noindent \texttt{ 7.0}
+
+\noindent \texttt{12.0}
+
+\noindent \texttt{40.0}
+
+
+\subsubsection{ To read the output files }
+
+The output format is :
+
+{\small\texttt{date, time-step number, section number, section name, section slope coefficient, class number,
+class name, class bound 1 , classe bound2, transport\_direction1 , transport\_direction2, transport\_total}}\\
+
+
+For sections with classes, the first \texttt{nclass-1} lines correspond to the transport for each class
+and the last line corresponds to the total transport summed over all classes. For sections with no classes, class number
+\texttt{ 1 } corresponds to \texttt{ total class } and this class is called \texttt{N}, meaning \texttt{none}.\\
+
+
+\noindent \texttt{ transport\_direction1 } is the positive part of the transport ( \texttt{ > = 0 } ).
+
+\noindent \texttt{ transport\_direction2 } is the negative part of the transport ( \texttt{ < = 0 } ).\\
+
+
+\noindent The \texttt{section slope coefficient} gives information about the significance of transports signs and direction:\\
+
+
+
+\begin{tabular}{|c|c|c|c|p{4cm}|}
+\hline
+section slope coefficient & section type & direction 1 & direction 2 & total transport \\ \hline
+0. & horizontal & northward & southward & postive: northward \\ \hline
+1000. & vertical & eastward & westward & postive: eastward \\ \hline
+\texttt{=/0, =/ 1000.} & diagonal & eastward & westward & postive: eastward \\ \hline
+\end{tabular}
+
+
+
+% -------------------------------------------------------------------------------------------------------------
+% Other Diagnostics
+% -------------------------------------------------------------------------------------------------------------
+\section{Other Diagnostics (\key{diahth}, \key{diaar5})}
+\label{DIA_diag_others}
+
+
+Aside from the standard model variables, other diagnostics can be computed
+on-line. The available ready-to-add diagnostics routines can be found in directory DIA.
+Among the available diagnostics the following ones are obtained when defining
+the \key{diahth} CPP key:
+
+- the mixed layer depth (based on a density criterion, \citet{de_Boyer_Montegut_al_JGR04}) (\mdl{diahth})
+
+- the turbocline depth (based on a turbulent mixing coefficient criterion) (\mdl{diahth})
+
+- the depth of the 20\deg C isotherm (\mdl{diahth})
+
+- the depth of the thermocline (maximum of the vertical temperature gradient) (\mdl{diahth})
+
+The poleward heat and salt transports, their advective and diffusive component, and
+the meriodional stream function can be computed on-line in \mdl{diaptr}
+\np{ln\_diaptr} to true (see the \textit{\ngn{namptr} } namelist below).
+When \np{ln\_subbas}~=~true, transports and stream function are computed
+for the Atlantic, Indian, Pacific and Indo-Pacific Oceans (defined north of 30\deg S)
+as well as for the World Ocean. The sub-basin decomposition requires an input file
+(\ifile{subbasins}) which contains three 2D mask arrays, the Indo-Pacific mask
+been deduced from the sum of the Indian and Pacific mask (Fig~\ref{Fig_mask_subasins}).
+
+%------------------------------------------namptr----------------------------------------------------
+\namdisplay{namptr}
+%-------------------------------------------------------------------------------------------------------------
+%>>>>>>>>>>>>>>>>>>>>>>>>>>>>
+\begin{figure}[!t] \begin{center}
+\includegraphics[width=1.0\textwidth]{./TexFiles/Figures/Fig_mask_subasins.pdf}
+\caption{ \label{Fig_mask_subasins}
+Decomposition of the World Ocean (here ORCA2) into sub-basin used in to compute
+the heat and salt transports as well as the meridional stream-function: Atlantic basin (red),
+Pacific basin (green), Indian basin (bleue), Indo-Pacific basin (bleue+green).
+Note that semi-enclosed seas (Red, Med and Baltic seas) as well as Hudson Bay
+are removed from the sub-basins. Note also that the Arctic Ocean has been split
+into Atlantic and Pacific basins along the North fold line. }
+\end{center} \end{figure}
+%>>>>>>>>>>>>>>>>>>>>>>>>>>>>
+
+In addition, a series of diagnostics has been added in the \mdl{diaar5}.
+They corresponds to outputs that are required for AR5 simulations
+(see Section \ref{DIA_steric} below for one of them).
+Activating those outputs requires to define the \key{diaar5} CPP key.
+\\
+\\
+
+
+
+% ================================================================
+% Steric effect in sea surface height
+% ================================================================
+\section{Diagnosing the Steric effect in sea surface height}
+\label{DIA_steric}
+
+
+Changes in steric sea level are caused when changes in the density of the water
+column imply an expansion or contraction of the column. It is essentially produced
+through surface heating/cooling and to a lesser extent through non-linear effects of
+the equation of state (cabbeling, thermobaricity...).
+Non-Boussinesq models contain all ocean effects within the ocean acting
+on the sea level. In particular, they include the steric effect. In contrast,
+Boussinesq models, such as \NEMO, conserve volume, rather than mass,
+and so do not properly represent expansion or contraction. The steric effect is
+therefore not explicitely represented.
+This approximation does not represent a serious error with respect to the flow field
+calculated by the model \citep{Greatbatch_JGR94}, but extra attention is required
+when investigating sea level, as steric changes are an important
+contribution to local changes in sea level on seasonal and climatic time scales.
+This is especially true for investigation into sea level rise due to global warming.
+
+Fortunately, the steric contribution to the sea level consists of a spatially uniform
+component that can be diagnosed by considering the mass budget of the world
+ocean \citep{Greatbatch_JGR94}.
+In order to better understand how global mean sea level evolves and thus how
+the steric sea level can be diagnosed, we compare, in the following, the
+non-Boussinesq and Boussinesq cases.
+
+Let denote
+$\mathcal{M}$ the total mass of liquid seawater ($\mathcal{M}=\int_D \rho dv$),
+$\mathcal{V}$ the total volume of seawater ($\mathcal{V}=\int_D dv$),
+$\mathcal{A}$ the total surface of the ocean ($\mathcal{A}=\int_S ds$),
+$\bar{\rho}$ the global mean seawater (\textit{in situ}) density ($\bar{\rho}= 1/\mathcal{V} \int_D \rho \,dv$), and
+$\bar{\eta}$ the global mean sea level ($\bar{\eta}=1/\mathcal{A}\int_S \eta \,ds$).
+
+A non-Boussinesq fluid conserves mass. It satisfies the following relations:
+\begin{equation} \label{Eq_MV_nBq}
+\begin{split}
+\mathcal{M} &= \mathcal{V} \;\bar{\rho} \\
+\mathcal{V} &= \mathcal{A} \;\bar{\eta}
+\end{split}
+\end{equation}
+Temporal changes in total mass is obtained from the density conservation equation :
+\begin{equation} \label{Eq_Co_nBq}
+\frac{1}{e_3} \partial_t ( e_3\,\rho) + \nabla( \rho \, \textbf{U} ) = \left. \frac{\textit{emp}}{e_3}\right|_\textit{surface}
+\end{equation}
+where $\rho$ is the \textit{in situ} density, and \textit{emp} the surface mass
+exchanges with the other media of the Earth system (atmosphere, sea-ice, land).
+Its global averaged leads to the total mass change
+\begin{equation} \label{Eq_Mass_nBq}
+\partial_t \mathcal{M} = \mathcal{A} \;\overline{\textit{emp}}
+\end{equation}
+where $\overline{\textit{emp}}=\int_S \textit{emp}\,ds$ is the net mass flux
+through the ocean surface.
+Bringing \eqref{Eq_Mass_nBq} and the time derivative of \eqref{Eq_MV_nBq}
+together leads to the evolution equation of the mean sea level
+\begin{equation} \label{Eq_ssh_nBq}
+ \partial_t \bar{\eta} = \frac{\overline{\textit{emp}}}{ \bar{\rho}}
+ - \frac{\mathcal{V}}{\mathcal{A}} \;\frac{\partial_t \bar{\rho} }{\bar{\rho}}
+\end{equation}
+The first term in equation \eqref{Eq_ssh_nBq} alters sea level by adding or
+subtracting mass from the ocean.
+The second term arises from temporal changes in the global mean
+density; $i.e.$ from steric effects.
+
+In a Boussinesq fluid, $\rho$ is replaced by $\rho_o$ in all the equation except when $\rho$
+appears multiplied by the gravity ($i.e.$ in the hydrostatic balance of the primitive Equations).
+In particular, the mass conservation equation, \eqref{Eq_Co_nBq}, degenerates into
+the incompressibility equation:
+\begin{equation} \label{Eq_Co_Bq}
+\frac{1}{e_3} \partial_t ( e_3 ) + \nabla( \textbf{U} ) = \left. \frac{\textit{emp}}{\rho_o \,e_3}\right|_ \textit{surface}
+\end{equation}
+and the global average of this equation now gives the temporal change of the total volume,
+\begin{equation} \label{Eq_V_Bq}
+ \partial_t \mathcal{V} = \mathcal{A} \;\frac{\overline{\textit{emp}}}{\rho_o}
+\end{equation}
+Only the volume is conserved, not mass, or, more precisely, the mass which is conserved is the
+Boussinesq mass, $\mathcal{M}_o = \rho_o \mathcal{V}$. The total volume (or equivalently
+the global mean sea level) is altered only by net volume fluxes across the ocean surface,
+not by changes in mean mass of the ocean: the steric effect is missing in a Boussinesq fluid.
+
+Nevertheless, following \citep{Greatbatch_JGR94}, the steric effect on the volume can be
+diagnosed by considering the mass budget of the ocean.
+The apparent changes in $\mathcal{M}$, mass of the ocean, which are not induced by surface
+mass flux must be compensated by a spatially uniform change in the mean sea level due to
+expansion/contraction of the ocean \citep{Greatbatch_JGR94}. In others words, the Boussinesq
+mass, $\mathcal{M}_o$, can be related to $\mathcal{M}$, the total mass of the ocean seen
+by the Boussinesq model, via the steric contribution to the sea level, $\eta_s$, a spatially
+uniform variable, as follows:
+\begin{equation} \label{Eq_M_Bq}
+ \mathcal{M}_o = \mathcal{M} + \rho_o \,\eta_s \,\mathcal{A}
+\end{equation}
+Any change in $\mathcal{M}$ which cannot be explained by the net mass flux through
+the ocean surface is converted into a mean change in sea level. Introducing the total density
+anomaly, $\mathcal{D}= \int_D d_a \,dv$, where $d_a= (\rho -\rho_o ) / \rho_o$
+is the density anomaly used in \NEMO (cf. \S\ref{TRA_eos}) in \eqref{Eq_M_Bq}
+leads to a very simple form for the steric height:
+\begin{equation} \label{Eq_steric_Bq}
+ \eta_s = - \frac{1}{\mathcal{A}} \mathcal{D}
+\end{equation}
+
+The above formulation of the steric height of a Boussinesq ocean requires four remarks.
+First, one can be tempted to define $\rho_o$ as the initial value of $\mathcal{M}/\mathcal{V}$,
+$i.e.$ set $\mathcal{D}_{t=0}=0$, so that the initial steric height is zero. We do not
+recommend that. Indeed, in this case $\rho_o$ depends on the initial state of the ocean.
+Since $\rho_o$ has a direct effect on the dynamics of the ocean (it appears in the pressure
+gradient term of the momentum equation) it is definitively not a good idea when
+inter-comparing experiments.
+We better recommend to fixe once for all $\rho_o$ to $1035\;Kg\,m^{-3}$. This value is a
+sensible choice for the reference density used in a Boussinesq ocean climate model since,
+with the exception of only a small percentage of the ocean, density in the World Ocean
+varies by no more than 2$\%$ from this value (\cite{Gill1982}, page 47).
+
+Second, we have assumed here that the total ocean surface, $\mathcal{A}$, does not
+change when the sea level is changing as it is the case in all global ocean GCMs
+(wetting and drying of grid point is not allowed).
+
+Third, the discretisation of \eqref{Eq_steric_Bq} depends on the type of free surface
+which is considered. In the non linear free surface case, $i.e.$ \key{vvl} defined, it is
+given by
+\begin{equation} \label{Eq_discrete_steric_Bq}
+ \eta_s = - \frac{ \sum_{i,\,j,\,k} d_a\; e_{1t} e_{2t} e_{3t} }
+ { \sum_{i,\,j,\,k} e_{1t} e_{2t} e_{3t} }
+\end{equation}
+whereas in the linear free surface, the volume above the \textit{z=0} surface must be explicitly taken
+into account to better approximate the total ocean mass and thus the steric sea level:
+\begin{equation} \label{Eq_discrete_steric_Bq}
+ \eta_s = - \frac{ \sum_{i,\,j,\,k} d_a\; e_{1t}e_{2t}e_{3t} + \sum_{i,\,j} d_a\; e_{1t}e_{2t} \eta }
+ {\sum_{i,\,j,\,k} e_{1t}e_{2t}e_{3t} + \sum_{i,\,j} e_{1t}e_{2t} \eta }
+\end{equation}
+
+The fourth and last remark concerns the effective sea level and the presence of sea-ice.
+In the real ocean, sea ice (and snow above it) depresses the liquid seawater through
+its mass loading. This depression is a result of the mass of sea ice/snow system acting
+on the liquid ocean. There is, however, no dynamical effect associated with these depressions
+in the liquid ocean sea level, so that there are no associated ocean currents. Hence, the
+dynamically relevant sea level is the effective sea level, $i.e.$ the sea level as if sea ice
+(and snow) were converted to liquid seawater \citep{Campin_al_OM08}. However,
+in the current version of \NEMO the sea-ice is levitating above the ocean without
+mass exchanges between ice and ocean. Therefore the model effective sea level
+is always given by $\eta + \eta_s$, whether or not there is sea ice present.
+
+In AR5 outputs, the thermosteric sea level is demanded. It is steric sea level due to
+changes in ocean density arising just from changes in temperature. It is given by:
+\begin{equation} \label{Eq_thermosteric_Bq}
+ \eta_s = - \frac{1}{\mathcal{A}} \int_D d_a(T,S_o,p_o) \,dv
+\end{equation}
+where $S_o$ and $p_o$ are the initial salinity and pressure, respectively.
+
+Both steric and thermosteric sea level are computed in \mdl{diaar5} which needs
+the \key{diaar5} defined to be called.
+
+% ================================================================
+
+
+
+
+
+
+
+
+
+
+
+
Index: /branches/UKMO/dev_r5518_GO6_under_ice_relax_dr_hook/DOC/TexFiles/Chapters/Chap_DOM.tex
===================================================================
--- /branches/UKMO/dev_r5518_GO6_under_ice_relax_dr_hook/DOC/TexFiles/Chapters/Chap_DOM.tex (revision 11643)
+++ /branches/UKMO/dev_r5518_GO6_under_ice_relax_dr_hook/DOC/TexFiles/Chapters/Chap_DOM.tex (revision 11643)
@@ -0,0 +1,911 @@
+% ================================================================
+% Chapter 2 � Space and Time Domain (DOM)
+% ================================================================
+\chapter{Space Domain (DOM) }
+\label{DOM}
+\minitoc
+
+% Missing things:
+% - istate: description of the initial state ==> this has to be put elsewhere..
+% perhaps in MISC ? By the way the initialisation of T S and dynamics
+% should be put outside of DOM routine (better with TRC staff and off-line
+% tracers)
+% -geo2ocean: how to switch from geographic to mesh coordinate
+% - domclo: closed sea and lakes.... management of closea sea area : specific to global configuration, both forced and coupled
+
+
+\newpage
+$\ $\newline % force a new ligne
+
+Having defined the continuous equations in Chap.~\ref{PE} and chosen a time
+discretization Chap.~\ref{STP}, we need to choose a discretization on a grid,
+and numerical algorithms. In the present chapter, we provide a general description
+of the staggered grid used in \NEMO, and other information relevant to the main
+directory routines as well as the DOM (DOMain) directory.
+
+$\ $\newline % force a new lign
+
+% ================================================================
+% Fundamentals of the Discretisation
+% ================================================================
+\section{Fundamentals of the Discretisation}
+\label{DOM_basics}
+
+% -------------------------------------------------------------------------------------------------------------
+% Arrangement of Variables
+% -------------------------------------------------------------------------------------------------------------
+\subsection{Arrangement of Variables}
+\label{DOM_cell}
+
+%>>>>>>>>>>>>>>>>>>>>>>>>>>>>
+\begin{figure}[!tb] \begin{center}
+\includegraphics[width=0.90\textwidth]{./TexFiles/Figures/Fig_cell.pdf}
+\caption{ \label{Fig_cell}
+Arrangement of variables. $t$ indicates scalar points where temperature,
+salinity, density, pressure and horizontal divergence are defined. ($u$,$v$,$w$)
+indicates vector points, and $f$ indicates vorticity points where both relative and
+planetary vorticities are defined}
+\end{center} \end{figure}
+%>>>>>>>>>>>>>>>>>>>>>>>>>>>>
+
+The numerical techniques used to solve the Primitive Equations in this model are
+based on the traditional, centred second-order finite difference approximation.
+Special attention has been given to the homogeneity of the solution in the three
+space directions. The arrangement of variables is the same in all directions.
+It consists of cells centred on scalar points ($t$, $S$, $p$, $\rho$) with vector
+points $(u, v, w)$ defined in the centre of each face of the cells (Fig. \ref{Fig_cell}).
+This is the generalisation to three dimensions of the well-known ``C'' grid in
+Arakawa's classification \citep{Mesinger_Arakawa_Bk76}. The relative and
+planetary vorticity, $\zeta$ and $f$, are defined in the centre of each vertical edge
+and the barotropic stream function $\psi$ is defined at horizontal points overlying
+the $\zeta$ and $f$-points.
+
+The ocean mesh ($i.e.$ the position of all the scalar and vector points) is defined
+by the transformation that gives ($\lambda$ ,$\varphi$ ,$z$) as a function of $(i,j,k)$.
+The grid-points are located at integer or integer and a half value of $(i,j,k)$ as
+indicated on Table \ref{Tab_cell}. In all the following, subscripts $u$, $v$, $w$,
+$f$, $uw$, $vw$ or $fw$ indicate the position of the grid-point where the scale
+factors are defined. Each scale factor is defined as the local analytical value
+provided by \eqref{Eq_scale_factors}. As a result, the mesh on which partial
+derivatives $\frac{\partial}{\partial \lambda}, \frac{\partial}{\partial \varphi}$, and
+$\frac{\partial}{\partial z} $ are evaluated is a uniform mesh with a grid size of unity.
+Discrete partial derivatives are formulated by the traditional, centred second order
+finite difference approximation while the scale factors are chosen equal to their
+local analytical value. An important point here is that the partial derivative of the
+scale factors must be evaluated by centred finite difference approximation, not
+from their analytical expression. This preserves the symmetry of the discrete set
+of equations and therefore satisfies many of the continuous properties (see
+Appendix~\ref{Apdx_C}). A similar, related remark can be made about the domain
+size: when needed, an area, volume, or the total ocean depth must be evaluated
+as the sum of the relevant scale factors (see \eqref{DOM_bar}) in the next section).
+
+%>>>>>>>>>>>>>>>>>>>>>>>>>>>>
+\begin{table}[!tb]
+\begin{center} \begin{tabular}{|p{46pt}|p{56pt}|p{56pt}|p{56pt}|}
+\hline
+T &$i$ & $j$ & $k$ \\ \hline
+u & $i+1/2$ & $j$ & $k$ \\ \hline
+v & $i$ & $j+1/2$ & $k$ \\ \hline
+w & $i$ & $j$ & $k+1/2$ \\ \hline
+f & $i+1/2$ & $j+1/2$ & $k$ \\ \hline
+uw & $i+1/2$ & $j$ & $k+1/2$ \\ \hline
+vw & $i$ & $j+1/2$ & $k+1/2$ \\ \hline
+fw & $i+1/2$ & $j+1/2$ & $k+1/2$ \\ \hline
+\end{tabular}
+\caption{ \label{Tab_cell}
+Location of grid-points as a function of integer or integer and a half value of the column,
+line or level. This indexing is only used for the writing of the semi-discrete equation.
+In the code, the indexing uses integer values only and has a reverse direction
+in the vertical (see \S\ref{DOM_Num_Index})}
+\end{center}
+\end{table}
+%>>>>>>>>>>>>>>>>>>>>>>>>>>>>
+
+% -------------------------------------------------------------------------------------------------------------
+% Vector Invariant Formulation
+% -------------------------------------------------------------------------------------------------------------
+\subsection{Discrete Operators}
+\label{DOM_operators}
+
+Given the values of a variable $q$ at adjacent points, the differencing and
+averaging operators at the midpoint between them are:
+\begin{subequations} \label{Eq_di_mi}
+\begin{align}
+ \delta _i [q] &= \ \ q(i+1/2) - q(i-1/2) \\
+ \overline q^{\,i} &= \left\{ q(i+1/2) + q(i-1/2) \right\} \; / \; 2
+\end{align}
+\end{subequations}
+
+Similar operators are defined with respect to $i+1/2$, $j$, $j+1/2$, $k$, and
+$k+1/2$. Following \eqref{Eq_PE_grad} and \eqref{Eq_PE_lap}, the gradient of a
+variable $q$ defined at a $t$-point has its three components defined at $u$-, $v$-
+and $w$-points while its Laplacien is defined at $t$-point. These operators have
+the following discrete forms in the curvilinear $s$-coordinate system:
+\begin{equation} \label{Eq_DOM_grad}
+\nabla q\equiv \frac{1}{e_{1u} } \delta _{i+1/2 } [q] \;\,\mathbf{i}
+ + \frac{1}{e_{2v} } \delta _{j+1/2 } [q] \;\,\mathbf{j}
+ + \frac{1}{e_{3w}} \delta _{k+1/2} [q] \;\,\mathbf{k}
+\end{equation}
+\begin{multline} \label{Eq_DOM_lap}
+\Delta q\equiv \frac{1}{e_{1t}\,e_{2t}\,e_{3t} }
+ \;\left( \delta_i \left[ \frac{e_{2u}\,e_{3u}} {e_{1u}} \;\delta_{i+1/2} [q] \right]
++ \delta_j \left[ \frac{e_{1v}\,e_{3v}} {e_{2v}} \;\delta_{j+1/2} [q] \right] \; \right) \\
++\frac{1}{e_{3t}} \delta_k \left[ \frac{1}{e_{3w} } \;\delta_{k+1/2} [q] \right]
+\end{multline}
+
+Following \eqref{Eq_PE_curl} and \eqref{Eq_PE_div}, a vector ${\rm {\bf A}}=\left( a_1,a_2,a_3\right)$
+defined at vector points $(u,v,w)$ has its three curl components defined at $vw$-, $uw$,
+and $f$-points, and its divergence defined at $t$-points:
+\begin{eqnarray} \label{Eq_DOM_curl}
+ \nabla \times {\rm {\bf A}}\equiv &
+ \frac{1}{e_{2v}\,e_{3vw} } \ \left( \delta_{j +1/2} \left[e_{3w}\,a_3 \right] -\delta_{k+1/2} \left[e_{2v} \,a_2 \right] \right) &\ \mathbf{i} \\
+ +& \frac{1}{e_{2u}\,e_{3uw}} \ \left( \delta_{k+1/2} \left[e_{1u}\,a_1 \right] -\delta_{i +1/2} \left[e_{3w}\,a_3 \right] \right) &\ \mathbf{j} \\
+ +& \frac{1}{e_{1f} \,e_{2f} } \ \left( \delta_{i +1/2} \left[e_{2v}\,a_2 \right] -\delta_{j +1/2} \left[e_{1u}\,a_1 \right] \right) &\ \mathbf{k}
+ \end{eqnarray}
+\begin{equation} \label{Eq_DOM_div}
+\nabla \cdot \rm{\bf A}=\frac{1}{e_{1t}\,e_{2t}\,e_{3t}}\left( \delta_i \left[e_{2u}\,e_{3u}\,a_1 \right]
+ +\delta_j \left[e_{1v}\,e_{3v}\,a_2 \right] \right)+\frac{1}{e_{3t} }\delta_k \left[a_3 \right]
+\end{equation}
+
+In the special case of a pure $z$-coordinate system, \eqref{Eq_DOM_lap} and
+\eqref{Eq_DOM_div} can be simplified. In this case, the vertical scale factor
+becomes a function of the single variable $k$ and thus does not depend on the
+horizontal location of a grid point. For example \eqref{Eq_DOM_div} reduces to:
+\begin{equation*}
+\nabla \cdot \rm{\bf A}=\frac{1}{e_{1t}\,e_{2t}} \left( \delta_i \left[e_{2u}\,a_1 \right]
+ +\delta_j \left[e_{1v}\, a_2 \right] \right)
+ +\frac{1}{e_{3t}} \delta_k \left[ a_3 \right]
+\end{equation*}
+
+The vertical average over the whole water column denoted by an overbar becomes
+for a quantity $q$ which is a masked field (i.e. equal to zero inside solid area):
+\begin{equation} \label{DOM_bar}
+\bar q = \frac{1}{H} \int_{k^b}^{k^o} {q\;e_{3q} \,dk}
+ \equiv \frac{1}{H_q }\sum\limits_k {q\;e_{3q} }
+\end{equation}
+where $H_q$ is the ocean depth, which is the masked sum of the vertical scale
+factors at $q$ points, $k^b$ and $k^o$ are the bottom and surface $k$-indices,
+and the symbol $k^o$ refers to a summation over all grid points of the same type
+in the direction indicated by the subscript (here $k$).
+
+In continuous form, the following properties are satisfied:
+\begin{equation} \label{Eq_DOM_curl_grad}
+\nabla \times \nabla q ={\rm {\bf {0}}}
+\end{equation}
+\begin{equation} \label{Eq_DOM_div_curl}
+\nabla \cdot \left( {\nabla \times {\rm {\bf A}}} \right)=0
+\end{equation}
+
+It is straightforward to demonstrate that these properties are verified locally in
+discrete form as soon as the scalar $q$ is taken at $t$-points and the vector
+\textbf{A} has its components defined at vector points $(u,v,w)$.
+
+Let $a$ and $b$ be two fields defined on the mesh, with value zero inside
+continental area. Using integration by parts it can be shown that the differencing
+operators ($\delta_i$, $\delta_j$ and $\delta_k$) are anti-symmetric linear
+operators, and further that the averaging operators $\overline{\,\cdot\,}^{\,i}$,
+$\overline{\,\cdot\,}^{\,k}$ and $\overline{\,\cdot\,}^{\,k}$) are symmetric linear
+operators, $i.e.$
+\begin{align}
+\label{DOM_di_adj}
+\sum\limits_i { a_i \;\delta _i \left[ b \right]}
+ &\equiv -\sum\limits_i {\delta _{i+1/2} \left[ a \right]\;b_{i+1/2} } \\
+\label{DOM_mi_adj}
+\sum\limits_i { a_i \;\overline b^{\,i}}
+ & \equiv \quad \sum\limits_i {\overline a ^{\,i+1/2}\;b_{i+1/2} }
+\end{align}
+
+In other words, the adjoint of the differencing and averaging operators are
+$\delta_i^*=\delta_{i+1/2}$ and
+${(\overline{\,\cdot \,}^{\,i})}^*= \overline{\,\cdot\,}^{\,i+1/2}$, respectively.
+These two properties will be used extensively in the Appendix~\ref{Apdx_C} to
+demonstrate integral conservative properties of the discrete formulation chosen.
+
+% -------------------------------------------------------------------------------------------------------------
+% Numerical Indexing
+% -------------------------------------------------------------------------------------------------------------
+\subsection{Numerical Indexing}
+\label{DOM_Num_Index}
+
+%>>>>>>>>>>>>>>>>>>>>>>>>>>>>
+\begin{figure}[!tb] \begin{center}
+\includegraphics[width=0.90\textwidth]{./TexFiles/Figures/Fig_index_hor.pdf}
+\caption{ \label{Fig_index_hor}
+Horizontal integer indexing used in the \textsc{Fortran} code. The dashed area indicates
+the cell in which variables contained in arrays have the same $i$- and $j$-indices}
+\end{center} \end{figure}
+%>>>>>>>>>>>>>>>>>>>>>>>>>>>>
+
+The array representation used in the \textsc{Fortran} code requires an integer
+indexing while the analytical definition of the mesh (see \S\ref{DOM_cell}) is
+associated with the use of integer values for $t$-points and both integer and
+integer and a half values for all the other points. Therefore a specific integer
+indexing must be defined for points other than $t$-points ($i.e.$ velocity and
+vorticity grid-points). Furthermore, the direction of the vertical indexing has
+been changed so that the surface level is at $k=1$.
+
+% -----------------------------------
+% Horizontal Indexing
+% -----------------------------------
+\subsubsection{Horizontal Indexing}
+\label{DOM_Num_Index_hor}
+
+The indexing in the horizontal plane has been chosen as shown in Fig.\ref{Fig_index_hor}.
+For an increasing $i$ index ($j$ index), the $t$-point and the eastward $u$-point
+(northward $v$-point) have the same index (see the dashed area in Fig.\ref{Fig_index_hor}).
+A $t$-point and its nearest northeast $f$-point have the same $i$-and $j$-indices.
+
+% -----------------------------------
+% Vertical indexing
+% -----------------------------------
+\subsubsection{Vertical Indexing}
+\label{DOM_Num_Index_vertical}
+
+In the vertical, the chosen indexing requires special attention since the
+$k$-axis is re-orientated downward in the \textsc{Fortran} code compared
+to the indexing used in the semi-discrete equations and given in \S\ref{DOM_cell}.
+The sea surface corresponds to the $w$-level $k=1$ which is the same index
+as $t$-level just below (Fig.\ref{Fig_index_vert}). The last $w$-level ($k=jpk$)
+either corresponds to the ocean floor or is inside the bathymetry while the last
+$t$-level is always inside the bathymetry (Fig.\ref{Fig_index_vert}). Note that
+for an increasing $k$ index, a $w$-point and the $t$-point just below have the
+same $k$ index, in opposition to what is done in the horizontal plane where
+it is the $t$-point and the nearest velocity points in the direction of the horizontal
+axis that have the same $i$ or $j$ index (compare the dashed area in
+Fig.\ref{Fig_index_hor} and \ref{Fig_index_vert}). Since the scale factors are
+chosen to be strictly positive, a \emph{minus sign} appears in the \textsc{Fortran}
+code \emph{before all the vertical derivatives} of the discrete equations given in
+this documentation.
+
+%>>>>>>>>>>>>>>>>>>>>>>>>>>>>
+\begin{figure}[!pt] \begin{center}
+\includegraphics[width=.90\textwidth]{./TexFiles/Figures/Fig_index_vert.pdf}
+\caption{ \label{Fig_index_vert}
+Vertical integer indexing used in the \textsc{Fortran } code. Note that
+the $k$-axis is orientated downward. The dashed area indicates the cell in
+which variables contained in arrays have the same $k$-index.}
+\end{center} \end{figure}
+%>>>>>>>>>>>>>>>>>>>>>>>>>>>>
+
+% -----------------------------------
+% Domain Size
+% -----------------------------------
+\subsubsection{Domain Size}
+\label{DOM_size}
+
+The total size of the computational domain is set by the parameters \np{jpiglo},
+\np{jpjglo} and \np{jpkdta} in the $i$, $j$ and $k$ directions respectively. They are
+given as namelist variables in the \ngn{namcfg} namelist.
+
+Note that are other namelist variables in the \ngn{namcfg} namelist that refer to
+ the domain size.
+The two variables \np{jpidta} and \np{jpjdta} may be larger than \np{jpiglo}, \np{jpjglo}
+when the user wants to use only a sub-region of a given configuration. This is
+the "zoom" capability described in \S\ref{MISC_zoom}. In most applications of
+the model, $jpidta=jpiglo$, $jpjdta=jpjglo$, and $jpizoom=jpjzoom=1$. Parameters
+$jpi$ and $jpj$ refer to the size of each processor subdomain when the code is
+run in parallel using domain decomposition (\key{mpp\_mpi} defined, see
+\S\ref{LBC_mpp}).
+
+
+$\ $\newline % force a new lign
+
+% ================================================================
+% Domain: Horizontal Grid (mesh)
+% ================================================================
+\section [Domain: Horizontal Grid (mesh) (\textit{domhgr})]
+ {Domain: Horizontal Grid (mesh) \small{(\mdl{domhgr} module)} }
+\label{DOM_hgr}
+
+% -------------------------------------------------------------------------------------------------------------
+% Coordinates and scale factors
+% -------------------------------------------------------------------------------------------------------------
+\subsection{Coordinates and scale factors}
+\label{DOM_hgr_coord_e}
+
+The ocean mesh ($i.e.$ the position of all the scalar and vector points) is defined
+by the transformation that gives $(\lambda,\varphi,z)$ as a function of $(i,j,k)$.
+The grid-points are located at integer or integer and a half values of as indicated
+in Table~\ref{Tab_cell}. The associated scale factors are defined using the
+analytical first derivative of the transformation \eqref{Eq_scale_factors}. These
+definitions are done in two modules, \mdl{domhgr} and \mdl{domzgr}, which
+provide the horizontal and vertical meshes, respectively. This section deals with
+the horizontal mesh parameters.
+
+In a horizontal plane, the location of all the model grid points is defined from the
+analytical expressions of the longitude $\lambda$ and latitude $\varphi$ as a
+function of $(i,j)$. The horizontal scale factors are calculated using
+\eqref{Eq_scale_factors}. For example, when the longitude and latitude are
+function of a single value ($i$ and $j$, respectively) (geographical configuration
+of the mesh), the horizontal mesh definition reduces to define the wanted
+$\lambda(i)$, $\varphi(j)$, and their derivatives $\lambda'(i)$ $\varphi'(j)$ in the
+\mdl{domhgr} module. The model computes the grid-point positions and scale
+factors in the horizontal plane as follows:
+\begin{flalign*}
+\lambda_t &\equiv \text{glamt}= \lambda(i) & \varphi_t &\equiv \text{gphit} = \varphi(j)\\
+\lambda_u &\equiv \text{glamu}= \lambda(i+1/2)& \varphi_u &\equiv \text{gphiu}= \varphi(j)\\
+\lambda_v &\equiv \text{glamv}= \lambda(i) & \varphi_v &\equiv \text{gphiv} = \varphi(j+1/2)\\
+\lambda_f &\equiv \text{glamf }= \lambda(i+1/2)& \varphi_f &\equiv \text{gphif }= \varphi(j+1/2)
+\end{flalign*}
+\begin{flalign*}
+e_{1t} &\equiv \text{e1t} = r_a |\lambda'(i) \; \cos\varphi(j) |&
+e_{2t} &\equiv \text{e2t} = r_a |\varphi'(j)| \\
+e_{1u} &\equiv \text{e1t} = r_a |\lambda'(i+1/2) \; \cos\varphi(j) |&
+e_{2u} &\equiv \text{e2t} = r_a |\varphi'(j)|\\
+e_{1v} &\equiv \text{e1t} = r_a |\lambda'(i) \; \cos\varphi(j+1/2) |&
+e_{2v} &\equiv \text{e2t} = r_a |\varphi'(j+1/2)|\\
+e_{1f} &\equiv \text{e1t} = r_a |\lambda'(i+1/2)\; \cos\varphi(j+1/2) |&
+e_{2f} &\equiv \text{e2t} = r_a |\varphi'(j+1/2)|
+\end{flalign*}
+where the last letter of each computational name indicates the grid point
+considered and $r_a$ is the earth radius (defined in \mdl{phycst} along with
+all universal constants). Note that the horizontal position of and scale factors
+at $w$-points are exactly equal to those of $t$-points, thus no specific arrays
+are defined at $w$-points.
+
+Note that the definition of the scale factors ($i.e.$ as the analytical first derivative
+of the transformation that gives $(\lambda,\varphi,z)$ as a function of $(i,j,k)$) is
+specific to the \NEMO model \citep{Marti_al_JGR92}. As an example, $e_{1t}$ is defined
+locally at a $t$-point, whereas many other models on a C grid choose to define
+such a scale factor as the distance between the $U$-points on each side of the
+$t$-point. Relying on an analytical transformation has two advantages: firstly, there
+is no ambiguity in the scale factors appearing in the discrete equations, since they
+are first introduced in the continuous equations; secondly, analytical transformations
+encourage good practice by the definition of smoothly varying grids (rather than
+allowing the user to set arbitrary jumps in thickness between adjacent layers)
+\citep{Treguier1996}. An example of the effect of such a choice is shown in
+Fig.~\ref{Fig_zgr_e3}.
+%>>>>>>>>>>>>>>>>>>>>>>>>>>>>
+\begin{figure}[!t] \begin{center}
+\includegraphics[width=0.90\textwidth]{./TexFiles/Figures/Fig_zgr_e3.pdf}
+\caption{ \label{Fig_zgr_e3}
+Comparison of (a) traditional definitions of grid-point position and grid-size in the vertical,
+and (b) analytically derived grid-point position and scale factors.
+For both grids here, the same $w$-point depth has been chosen but in (a) the
+$t$-points are set half way between $w$-points while in (b) they are defined from
+an analytical function: $z(k)=5\,(i-1/2)^3 - 45\,(i-1/2)^2 + 140\,(i-1/2) - 150$.
+Note the resulting difference between the value of the grid-size $\Delta_k$ and
+those of the scale factor $e_k$. }
+\end{center} \end{figure}
+%>>>>>>>>>>>>>>>>>>>>>>>>>>>>
+
+% -------------------------------------------------------------------------------------------------------------
+% Choice of horizontal grid
+% -------------------------------------------------------------------------------------------------------------
+\subsection{Choice of horizontal grid}
+\label{DOM_hgr_msh_choice}
+
+The user has three options available in defining a horizontal grid, which involve
+the namelist variable \np{jphgr\_mesh} of the \ngn{namcfg} namelist.
+\begin{description}
+\item[\np{jphgr\_mesh}=0] The most general curvilinear orthogonal grids.
+The coordinates and their first derivatives with respect to $i$ and $j$ are provided
+in a input file (\ifile{coordinates}), read in \rou{hgr\_read} subroutine of the domhgr module.
+\item[\np{jphgr\_mesh}=1 to 5] A few simple analytical grids are provided (see below).
+For other analytical grids, the \mdl{domhgr} module must be modified by the user.
+\end{description}
+
+There are two simple cases of geographical grids on the sphere. With
+\np{jphgr\_mesh}=1, the grid (expressed in degrees) is regular in space,
+with grid sizes specified by parameters \np{ppe1\_deg} and \np{ppe2\_deg},
+respectively. Such a geographical grid can be very anisotropic at high latitudes
+because of the convergence of meridians (the zonal scale factors $e_1$
+become much smaller than the meridional scale factors $e_2$). The Mercator
+grid (\np{jphgr\_mesh}=4) avoids this anisotropy by refining the meridional scale
+factors in the same way as the zonal ones. In this case, meridional scale factors
+and latitudes are calculated analytically using the formulae appropriate for
+a Mercator projection, based on \np{ppe1\_deg} which is a reference grid spacing
+at the equator (this applies even when the geographical equator is situated outside
+the model domain).
+%%%
+\gmcomment{ give here the analytical expression of the Mercator mesh}
+%%%
+In these two cases (\np{jphgr\_mesh}=1 or 4), the grid position is defined by the
+longitude and latitude of the south-westernmost point (\np{ppglamt0}
+and \np{ppgphi0}). Note that for the Mercator grid the user need only provide
+an approximate starting latitude: the real latitude will be recalculated analytically,
+in order to ensure that the equator corresponds to line passing through $t$-
+and $u$-points.
+
+Rectangular grids ignoring the spherical geometry are defined with
+\np{jphgr\_mesh} = 2, 3, 5. The domain is either an $f$-plane (\np{jphgr\_mesh} = 2,
+Coriolis factor is constant) or a beta-plane (\np{jphgr\_mesh} = 3, the Coriolis factor
+is linear in the $j$-direction). The grid size is uniform in meter in each direction,
+and given by the parameters \np{ppe1\_m} and \np{ppe2\_m} respectively.
+The zonal grid coordinate (\textit{glam} arrays) is in kilometers, starting at zero
+with the first $t$-point. The meridional coordinate (gphi. arrays) is in kilometers,
+and the second $t$-point corresponds to coordinate $gphit=0$. The input
+variable \np{ppglam0} is ignored. \np{ppgphi0} is used to set the reference
+latitude for computation of the Coriolis parameter. In the case of the beta plane,
+\np{ppgphi0} corresponds to the center of the domain. Finally, the special case
+\np{jphgr\_mesh}=5 corresponds to a beta plane in a rotated domain for the
+GYRE configuration, representing a classical mid-latitude double gyre system.
+The rotation allows us to maximize the jet length relative to the gyre areas
+(and the number of grid points).
+
+The choice of the grid must be consistent with the boundary conditions specified
+by the parameter \np{jperio} (see {\S\ref{LBC}).
+
+% -------------------------------------------------------------------------------------------------------------
+% Grid files
+% -------------------------------------------------------------------------------------------------------------
+\subsection{Output Grid files}
+\label{DOM_hgr_files}
+
+All the arrays relating to a particular ocean model configuration (grid-point
+position, scale factors, masks) can be saved in files if $\np{nn\_msh} \not= 0$
+(namelist variable in \ngn{namdom}). This can be particularly useful for plots and off-line
+diagnostics. In some cases, the user may choose to make a local modification
+of a scale factor in the code. This is the case in global configurations when
+restricting the width of a specific strait (usually a one-grid-point strait that
+happens to be too wide due to insufficient model resolution). An example
+is Gibraltar Strait in the ORCA2 configuration. When such modifications are done,
+the output grid written when $\np{nn\_msh} \not=0$ is no more equal to the input grid.
+
+$\ $\newline % force a new lign
+
+% ================================================================
+% Domain: Vertical Grid (domzgr)
+% ================================================================
+\section [Domain: Vertical Grid (\textit{domzgr})]
+ {Domain: Vertical Grid \small{(\mdl{domzgr} module)} }
+\label{DOM_zgr}
+%-----------------------------------------nam_zgr & namdom-------------------------------------------
+\namdisplay{namzgr}
+\namdisplay{namdom}
+%-------------------------------------------------------------------------------------------------------------
+
+Variables are defined through the \ngn{namzgr} and \ngn{namdom} namelists.
+In the vertical, the model mesh is determined by four things:
+(1) the bathymetry given in meters ;
+(2) the number of levels of the model (\jp{jpk}) ;
+(3) the analytical transformation $z(i,j,k)$ and the vertical scale factors
+(derivatives of the transformation) ;
+and (4) the masking system, $i.e.$ the number of wet model levels at each
+$(i,j)$ column of points.
+
+%>>>>>>>>>>>>>>>>>>>>>>>>>>>>
+\begin{figure}[!tb] \begin{center}
+\includegraphics[width=1.0\textwidth]{./TexFiles/Figures/Fig_z_zps_s_sps.pdf}
+\caption{ \label{Fig_z_zps_s_sps}
+The ocean bottom as seen by the model:
+(a) $z$-coordinate with full step,
+(b) $z$-coordinate with partial step,
+(c) $s$-coordinate: terrain following representation,
+(d) hybrid $s-z$ coordinate,
+(e) hybrid $s-z$ coordinate with partial step, and
+(f) same as (e) but with variable volume associated with the non-linear free surface.
+Note that the variable volume option (\key{vvl}) can be used with any of the
+5 coordinates (a) to (e).}
+\end{center} \end{figure}
+%>>>>>>>>>>>>>>>>>>>>>>>>>>>>
+
+The choice of a vertical coordinate, even if it is made through a namelist parameter,
+must be done once of all at the beginning of an experiment. It is not intended as an
+option which can be enabled or disabled in the middle of an experiment. Three main
+choices are offered (Fig.~\ref{Fig_z_zps_s_sps}a to c): $z$-coordinate with full step
+bathymetry (\np{ln\_zco}~=~true), $z$-coordinate with partial step bathymetry
+(\np{ln\_zps}~=~true), or generalized, $s$-coordinate (\np{ln\_sco}~=~true).
+Hybridation of the three main coordinates are available: $s-z$ or $s-zps$ coordinate
+(Fig.~\ref{Fig_z_zps_s_sps}d and \ref{Fig_z_zps_s_sps}e). When using the variable
+volume option \key{vvl} ($i.e.$ non-linear free surface), the coordinate follow the
+time-variation of the free surface so that the transformation is time dependent:
+$z(i,j,k,t)$ (Fig.~\ref{Fig_z_zps_s_sps}f). This option can be used with full step
+bathymetry or $s$-coordinate (hybrid and partial step coordinates have not
+yet been tested in NEMO v2.3). If using $z$-coordinate with partial step bathymetry
+(\np{ln\_zps}~=~true), ocean cavity beneath ice shelves can be open (\np{ln\_isfcav}~=~true).
+
+Contrary to the horizontal grid, the vertical grid is computed in the code and no
+provision is made for reading it from a file. The only input file is the bathymetry
+(in meters) (\ifile{bathy\_meter})
+\footnote{N.B. in full step $z$-coordinate, a \ifile{bathy\_level} file can replace the
+\ifile{bathy\_meter} file, so that the computation of the number of wet ocean point
+in each water column is by-passed}.
+After reading the bathymetry, the algorithm for vertical grid definition differs
+between the different options:
+\begin{description}
+\item[\textit{zco}] set a reference coordinate transformation $z_0 (k)$, and set $z(i,j,k,t)=z_0 (k)$.
+\item[\textit{zps}] set a reference coordinate transformation $z_0 (k)$, and
+calculate the thickness of the deepest level at each $(i,j)$ point using the
+bathymetry, to obtain the final three-dimensional depth and scale factor arrays.
+\item[\textit{sco}] smooth the bathymetry to fulfil the hydrostatic consistency
+criteria and set the three-dimensional transformation.
+\item[\textit{s-z} and \textit{s-zps}] smooth the bathymetry to fulfil the hydrostatic
+consistency criteria and set the three-dimensional transformation $z(i,j,k)$, and
+possibly introduce masking of extra land points to better fit the original bathymetry file
+\end{description}
+%%%
+\gmcomment{ add the description of the smoothing: envelop topography...}
+%%%
+
+The arrays describing the grid point depths and vertical scale factors
+are three dimensional arrays $(i,j,k)$ even in the case of $z$-coordinate with
+full step bottom topography. In non-linear free surface (\key{vvl}), their knowledge
+is required at \textit{before}, \textit{now} and \textit{after} time step, while they
+do not vary in time in linear free surface case.
+To improve the code readability while providing this flexibility, the vertical coordinate
+and scale factors are defined as functions of
+$(i,j,k)$ with "fs" as prefix (examples: \textit{fse3t\_b, fse3t\_n, fse3t\_a,}
+for the \textit{before}, \textit{now} and \textit{after} scale factors at $t$-point)
+that can be either three different arrays when \key{vvl} is defined, or a single fixed arrays.
+These functions are defined in the file \hf{domzgr\_substitute} of the DOM directory.
+They are used throughout the code, and replaced by the corresponding arrays at
+the time of pre-processing (CPP capability).
+
+% -------------------------------------------------------------------------------------------------------------
+% Meter Bathymetry
+% -------------------------------------------------------------------------------------------------------------
+\subsection{Meter Bathymetry}
+\label{DOM_bathy}
+
+Three options are possible for defining the bathymetry, according to the
+namelist variable \np{nn\_bathy}:
+\begin{description}
+\item[\np{nn\_bathy} = 0] a flat-bottom domain is defined. The total depth $z_w (jpk)$
+is given by the coordinate transformation. The domain can either be a closed
+basin or a periodic channel depending on the parameter \np{jperio}.
+\item[\np{nn\_bathy} = -1] a domain with a bump of topography one third of the
+domain width at the central latitude. This is meant for the "EEL-R5" configuration,
+a periodic or open boundary channel with a seamount.
+\item[\np{nn\_bathy} = 1] read a bathymetry. The \ifile{bathy\_meter} file (Netcdf format)
+provides the ocean depth (positive, in meters) at each grid point of the model grid.
+The bathymetry is usually built by interpolating a standard bathymetry product
+($e.g.$ ETOPO2) onto the horizontal ocean mesh. Defining the bathymetry also
+defines the coastline: where the bathymetry is zero, no model levels are defined
+(all levels are masked).
+\end{description}
+
+When a global ocean is coupled to an atmospheric model it is better to represent
+all large water bodies (e.g, great lakes, Caspian sea...) even if the model
+resolution does not allow their communication with the rest of the ocean.
+This is unnecessary when the ocean is forced by fixed atmospheric conditions,
+so these seas can be removed from the ocean domain. The user has the option
+to set the bathymetry in closed seas to zero (see \S\ref{MISC_closea}), but the
+code has to be adapted to the user's configuration.
+
+% -------------------------------------------------------------------------------------------------------------
+% z-coordinate and reference coordinate transformation
+% -------------------------------------------------------------------------------------------------------------
+\subsection[$z$-coordinate (\np{ln\_zco}]
+ {$z$-coordinate (\np{ln\_zco}=true) and reference coordinate}
+\label{DOM_zco}
+
+%>>>>>>>>>>>>>>>>>>>>>>>>>>>>
+\begin{figure}[!tb] \begin{center}
+\includegraphics[width=0.90\textwidth]{./TexFiles/Figures/Fig_zgr.pdf}
+\caption{ \label{Fig_zgr}
+Default vertical mesh for ORCA2: 30 ocean levels (L30). Vertical level functions for
+(a) T-point depth and (b) the associated scale factor as computed
+from \eqref{DOM_zgr_ana} using \eqref{DOM_zgr_coef} in $z$-coordinate.}
+\end{center} \end{figure}
+%>>>>>>>>>>>>>>>>>>>>>>>>>>>>
+
+The reference coordinate transformation $z_0 (k)$ defines the arrays $gdept_0$
+and $gdepw_0$ for $t$- and $w$-points, respectively. As indicated on
+Fig.\ref{Fig_index_vert} \jp{jpk} is the number of $w$-levels. $gdepw_0(1)$ is the
+ocean surface. There are at most \jp{jpk}-1 $t$-points inside the ocean, the
+additional $t$-point at $jk=jpk$ is below the sea floor and is not used.
+The vertical location of $w$- and $t$-levels is defined from the analytic expression
+of the depth $z_0(k)$ whose analytical derivative with respect to $k$ provides the
+vertical scale factors. The user must provide the analytical expression of both
+$z_0$ and its first derivative with respect to $k$. This is done in routine \mdl{domzgr}
+through statement functions, using parameters provided in the \ngn{namcfg} namelist.
+
+It is possible to define a simple regular vertical grid by giving zero stretching (\np{ppacr=0}).
+In that case, the parameters \jp{jpk} (number of $w$-levels) and \np{pphmax}
+(total ocean depth in meters) fully define the grid.
+
+For climate-related studies it is often desirable to concentrate the vertical resolution
+near the ocean surface. The following function is proposed as a standard for a
+$z$-coordinate (with either full or partial steps):
+\begin{equation} \label{DOM_zgr_ana}
+\begin{split}
+ z_0 (k) &= h_{sur} -h_0 \;k-\;h_1 \;\log \left[ {\,\cosh \left( {{(k-h_{th} )} / {h_{cr} }} \right)\,} \right] \\
+ e_3^0 (k) &= \left| -h_0 -h_1 \;\tanh \left( {{(k-h_{th} )} / {h_{cr} }} \right) \right|
+\end{split}
+\end{equation}
+where $k=1$ to \jp{jpk} for $w$-levels and $k=1$ to $k=1$ for $T-$levels. Such an
+expression allows us to define a nearly uniform vertical location of levels at the
+ocean top and bottom with a smooth hyperbolic tangent transition in between
+(Fig.~\ref{Fig_zgr}).
+
+The most used vertical grid for ORCA2 has $10~m$ ($500~m)$ resolution in the
+surface (bottom) layers and a depth which varies from 0 at the sea surface to a
+minimum of $-5000~m$. This leads to the following conditions:
+\begin{equation} \label{DOM_zgr_coef}
+\begin{split}
+ e_3 (1+1/2) &=10. \\
+ e_3 (jpk-1/2) &=500. \\
+ z(1) &=0. \\
+ z(jpk) &=-5000. \\
+\end{split}
+\end{equation}
+
+With the choice of the stretching $h_{cr} =3$ and the number of levels
+\jp{jpk}=$31$, the four coefficients $h_{sur}$, $h_{0}$, $h_{1}$, and $h_{th}$ in
+\eqref{DOM_zgr_ana} have been determined such that \eqref{DOM_zgr_coef} is
+satisfied, through an optimisation procedure using a bisection method. For the first
+standard ORCA2 vertical grid this led to the following values: $h_{sur} =4762.96$,
+$h_0 =255.58, h_1 =245.5813$, and $h_{th} =21.43336$. The resulting depths and
+scale factors as a function of the model levels are shown in Fig.~\ref{Fig_zgr} and
+given in Table \ref{Tab_orca_zgr}. Those values correspond to the parameters
+\np{ppsur}, \np{ppa0}, \np{ppa1}, \np{ppkth} in \ngn{namcfg} namelist.
+
+Rather than entering parameters $h_{sur}$, $h_{0}$, and $h_{1}$ directly, it is
+possible to recalculate them. In that case the user sets
+\np{ppsur}=\np{ppa0}=\np{ppa1}=999999., in \ngn{namcfg} namelist,
+and specifies instead the four following parameters:
+\begin{itemize}
+\item \np{ppacr}=$h_{cr} $: stretching factor (nondimensional). The larger
+\np{ppacr}, the smaller the stretching. Values from $3$ to $10$ are usual.
+\item \np{ppkth}=$h_{th} $: is approximately the model level at which maximum
+stretching occurs (nondimensional, usually of order 1/2 or 2/3 of \jp{jpk})
+\item \np{ppdzmin}: minimum thickness for the top layer (in meters)
+\item \np{pphmax}: total depth of the ocean (meters).
+\end{itemize}
+As an example, for the $45$ layers used in the DRAKKAR configuration those
+parameters are: \jp{jpk}=46, \np{ppacr}=9, \np{ppkth}=23.563, \np{ppdzmin}=6m,
+\np{pphmax}=5750m.
+
+%>>>>>>>>>>>>>>>>>>>>>>>>>>>>
+\begin{table} \begin{center} \begin{tabular}{c||r|r|r|r}
+\hline
+\textbf{LEVEL}& \textbf{gdept}& \textbf{gdepw}& \textbf{e3t }& \textbf{e3w } \\ \hline
+1 & \textbf{ 5.00} & 0.00 & \textbf{ 10.00} & 10.00 \\ \hline
+2 & \textbf{15.00} & 10.00 & \textbf{ 10.00} & 10.00 \\ \hline
+3 & \textbf{25.00} & 20.00 & \textbf{ 10.00} & 10.00 \\ \hline
+4 & \textbf{35.01} & 30.00 & \textbf{ 10.01} & 10.00 \\ \hline
+5 & \textbf{45.01} & 40.01 & \textbf{ 10.01} & 10.01 \\ \hline
+6 & \textbf{55.03} & 50.02 & \textbf{ 10.02} & 10.02 \\ \hline
+7 & \textbf{65.06} & 60.04 & \textbf{ 10.04} & 10.03 \\ \hline
+8 & \textbf{75.13} & 70.09 & \textbf{ 10.09} & 10.06 \\ \hline
+9 & \textbf{85.25} & 80.18 & \textbf{ 10.17} & 10.12 \\ \hline
+10 & \textbf{95.49} & 90.35 & \textbf{ 10.33} & 10.24 \\ \hline
+11 & \textbf{105.97} & 100.69 & \textbf{ 10.65} & 10.47 \\ \hline
+12 & \textbf{116.90} & 111.36 & \textbf{ 11.27} & 10.91 \\ \hline
+13 & \textbf{128.70} & 122.65 & \textbf{ 12.47} & 11.77 \\ \hline
+14 & \textbf{142.20} & 135.16 & \textbf{ 14.78} & 13.43 \\ \hline
+15 & \textbf{158.96} & 150.03 & \textbf{ 19.23} & 16.65 \\ \hline
+16 & \textbf{181.96} & 169.42 & \textbf{ 27.66} & 22.78 \\ \hline
+17 & \textbf{216.65} & 197.37 & \textbf{ 43.26} & 34.30 \\ \hline
+18 & \textbf{272.48} & 241.13 & \textbf{ 70.88} & 55.21 \\ \hline
+19 & \textbf{364.30} & 312.74 & \textbf{116.11} & 90.99 \\ \hline
+20 & \textbf{511.53} & 429.72 & \textbf{181.55} & 146.43 \\ \hline
+21 & \textbf{732.20} & 611.89 & \textbf{261.03} & 220.35 \\ \hline
+22 & \textbf{1033.22}& 872.87 & \textbf{339.39} & 301.42 \\ \hline
+23 & \textbf{1405.70}& 1211.59 & \textbf{402.26} & 373.31 \\ \hline
+24 & \textbf{1830.89}& 1612.98 & \textbf{444.87} & 426.00 \\ \hline
+25 & \textbf{2289.77}& 2057.13 & \textbf{470.55} & 459.47 \\ \hline
+26 & \textbf{2768.24}& 2527.22 & \textbf{484.95} & 478.83 \\ \hline
+27 & \textbf{3257.48}& 3011.90 & \textbf{492.70} & 489.44 \\ \hline
+28 & \textbf{3752.44}& 3504.46 & \textbf{496.78} & 495.07 \\ \hline
+29 & \textbf{4250.40}& 4001.16 & \textbf{498.90} & 498.02 \\ \hline
+30 & \textbf{4749.91}& 4500.02 & \textbf{500.00} & 499.54 \\ \hline
+31 & \textbf{5250.23}& 5000.00 & \textbf{500.56} & 500.33 \\ \hline
+\end{tabular} \end{center}
+\caption{ \label{Tab_orca_zgr}
+Default vertical mesh in $z$-coordinate for 30 layers ORCA2 configuration as computed
+from \eqref{DOM_zgr_ana} using the coefficients given in \eqref{DOM_zgr_coef}}
+\end{table}
+%>>>>>>>>>>>>>>>>>>>>>>>>>>>>
+
+% -------------------------------------------------------------------------------------------------------------
+% z-coordinate with partial step
+% -------------------------------------------------------------------------------------------------------------
+\subsection [$z$-coordinate with partial step (\np{ln\_zps})]
+ {$z$-coordinate with partial step (\np{ln\_zps}=.true.)}
+\label{DOM_zps}
+%--------------------------------------------namdom-------------------------------------------------------
+\namdisplay{namdom}
+%--------------------------------------------------------------------------------------------------------------
+
+In $z$-coordinate partial step, the depths of the model levels are defined by the
+reference analytical function $z_0 (k)$ as described in the previous
+section, \emph{except} in the bottom layer. The thickness of the bottom layer is
+allowed to vary as a function of geographical location $(\lambda,\varphi)$ to allow a
+better representation of the bathymetry, especially in the case of small
+slopes (where the bathymetry varies by less than one level thickness from
+one grid point to the next). The reference layer thicknesses $e_{3t}^0$ have been
+defined in the absence of bathymetry. With partial steps, layers from 1 to
+\jp{jpk}-2 can have a thickness smaller than $e_{3t}(jk)$. The model deepest layer (\jp{jpk}-1)
+is allowed to have either a smaller or larger thickness than $e_{3t}(jpk)$: the
+maximum thickness allowed is $2*e_{3t}(jpk-1)$. This has to be kept in mind when
+specifying values in \ngn{namdom} namelist, as the maximum depth \np{pphmax}
+in partial steps: for example, with
+\np{pphmax}$=5750~m$ for the DRAKKAR 45 layer grid, the maximum ocean depth
+allowed is actually $6000~m$ (the default thickness $e_{3t}(jpk-1)$ being $250~m$).
+Two variables in the namdom namelist are used to define the partial step
+vertical grid. The mimimum water thickness (in meters) allowed for a cell
+partially filled with bathymetry at level jk is the minimum of \np{rn\_e3zps\_min}
+(thickness in meters, usually $20~m$) or $e_{3t}(jk)*\np{rn\_e3zps\_rat}$ (a fraction,
+usually 10\%, of the default thickness $e_{3t}(jk)$).
+
+ \colorbox{yellow}{Add a figure here of pstep especially at last ocean level }
+
+% -------------------------------------------------------------------------------------------------------------
+% s-coordinate
+% -------------------------------------------------------------------------------------------------------------
+\subsection [$s$-coordinate (\np{ln\_sco})]
+ {$s$-coordinate (\np{ln\_sco}=true)}
+\label{DOM_sco}
+%------------------------------------------nam_zgr_sco---------------------------------------------------
+\namdisplay{namzgr_sco}
+%--------------------------------------------------------------------------------------------------------------
+Options are defined in \ngn{namzgr\_sco}.
+In $s$-coordinate (\np{ln\_sco}~=~true), the depth and thickness of the model
+levels are defined from the product of a depth field and either a stretching
+function or its derivative, respectively:
+
+\begin{equation} \label{DOM_sco_ana}
+\begin{split}
+ z(k) &= h(i,j) \; z_0(k) \\
+ e_3(k) &= h(i,j) \; z_0'(k)
+\end{split}
+\end{equation}
+
+where $h$ is the depth of the last $w$-level ($z_0(k)$) defined at the $t$-point
+location in the horizontal and $z_0(k)$ is a function which varies from $0$ at the sea
+surface to $1$ at the ocean bottom. The depth field $h$ is not necessary the ocean
+depth, since a mixed step-like and bottom-following representation of the
+topography can be used (Fig.~\ref{Fig_z_zps_s_sps}d-e) or an envelop bathymetry can be defined (Fig.~\ref{Fig_z_zps_s_sps}f).
+The namelist parameter \np{rn\_rmax} determines the slope at which the terrain-following coordinate intersects the sea bed and becomes a pseudo z-coordinate. The coordinate can also be hybridised by specifying \np{rn\_sbot\_min} and \np{rn\_sbot\_max} as the minimum and maximum depths at which the terrain-following vertical coordinate is calculated.
+
+Options for stretching the coordinate are provided as examples, but care must be taken to ensure that the vertical stretch used is appropriate for the application.
+
+The original default NEMO s-coordinate stretching is available if neither of the other options are specified as true (\np{ln\_sco\_SH94}~=~false and \np{ln\_sco\_SF12}~=~false.) This uses a depth independent $\tanh$ function for the stretching \citep{Madec_al_JPO96}:
+
+\begin{equation}
+ z = s_{min}+C\left(s\right)\left(H-s_{min}\right)
+ \label{eq:SH94_1}
+\end{equation}
+
+where $s_{min}$ is the depth at which the s-coordinate stretching starts and allows a z-coordinate to placed on top of the stretched coordinate, and z is the depth (negative down from the asea surface).
+
+\begin{equation}
+ s = -\frac{k}{n-1} \quad \text{ and } \quad 0 \leq k \leq n-1
+ \label{eq:s}
+\end{equation}
+
+\begin{equation} \label{DOM_sco_function}
+\begin{split}
+C(s) &= \frac{ \left[ \tanh{ \left( \theta \, (s+b) \right)}
+ - \tanh{ \left( \theta \, b \right)} \right]}
+ {2\;\sinh \left( \theta \right)}
+\end{split}
+\end{equation}
+
+A stretching function, modified from the commonly used \citet{Song_Haidvogel_JCP94} stretching (\np{ln\_sco\_SH94}~=~true), is also available and is more commonly used for shelf seas modelling:
+
+\begin{equation}
+ C\left(s\right) = \left(1 - b \right)\frac{ \sinh\left( \theta s\right)}{\sinh\left(\theta\right)} + \\
+ b\frac{ \tanh \left[ \theta \left(s + \frac{1}{2} \right)\right] - \tanh\left(\frac{\theta}{2}\right)}{ 2\tanh\left (\frac{\theta}{2}\right)}
+ \label{eq:SH94_2}
+\end{equation}
+
+%>>>>>>>>>>>>>>>>>>>>>>>>>>>>
+\begin{figure}[!ht] \begin{center}
+\includegraphics[width=1.0\textwidth]{./TexFiles/Figures/Fig_sco_function.pdf}
+\caption{ \label{Fig_sco_function}
+Examples of the stretching function applied to a seamount; from left to right:
+surface, surface and bottom, and bottom intensified resolutions}
+\end{center} \end{figure}
+%>>>>>>>>>>>>>>>>>>>>>>>>>>>>
+
+where $H_c$ is the critical depth (\np{rn\_hc}) at which the coordinate transitions from pure $\sigma$ to the stretched coordinate, and $\theta$ (\np{rn\_theta}) and $b$ (\np{rn\_bb}) are the surface and
+bottom control parameters such that $0\leqslant \theta \leqslant 20$, and
+$0\leqslant b\leqslant 1$. $b$ has been designed to allow surface and/or bottom
+increase of the vertical resolution (Fig.~\ref{Fig_sco_function}).
+
+Another example has been provided at version 3.5 (\np{ln\_sco\_SF12}) that allows a fixed surface resolution in an analytical terrain-following stretching \citet{Siddorn_Furner_OM12}. In this case the a stretching function $\gamma$ is defined such that:
+
+\begin{equation}
+z = -\gamma h \quad \text{ with } \quad 0 \leq \gamma \leq 1
+\label{eq:z}
+\end{equation}
+
+The function is defined with respect to $\sigma$, the unstretched terrain-following coordinate:
+
+\begin{equation} \label{DOM_gamma_deriv}
+\gamma= A\left(\sigma-\frac{1}{2}\left(\sigma^{2}+f\left(\sigma\right)\right)\right)+B\left(\sigma^{3}-f\left(\sigma\right)\right)+f\left(\sigma\right)
+\end{equation}
+
+Where:
+\begin{equation} \label{DOM_gamma}
+f\left(\sigma\right)=\left(\alpha+2\right)\sigma^{\alpha+1}-\left(\alpha+1\right)\sigma^{\alpha+2} \quad \text{ and } \quad \sigma = \frac{k}{n-1}
+\end{equation}
+
+This gives an analytical stretching of $\sigma$ that is solvable in $A$ and $B$ as a function of the user prescribed stretching parameter $\alpha$ (\np{rn\_alpha}) that stretches towards the surface ($\alpha > 1.0$) or the bottom ($\alpha < 1.0$) and user prescribed surface (\np{rn\_zs}) and bottom depths. The bottom cell depth in this example is given as a function of water depth:
+
+\begin{equation} \label{DOM_zb}
+Z_b= h a + b
+\end{equation}
+
+where the namelist parameters \np{rn\_zb\_a} and \np{rn\_zb\_b} are $a$ and $b$ respectively.
+
+%>>>>>>>>>>>>>>>>>>>>>>>>>>>>
+\begin{figure}[!ht]
+ \includegraphics[width=1.0\textwidth]{./TexFiles/Figures/FIG_DOM_compare_coordinates_surface.pdf}
+ \caption{A comparison of the \citet{Song_Haidvogel_JCP94} $S$-coordinate (solid lines), a 50 level $Z$-coordinate (contoured surfaces) and the \citet{Siddorn_Furner_OM12} $S$-coordinate (dashed lines) in the surface 100m for a idealised bathymetry that goes from 50m to 5500m depth. For clarity every third coordinate surface is shown.}
+ \label{fig_compare_coordinates_surface}
+\end{figure}
+%>>>>>>>>>>>>>>>>>>>>>>>>>>>>
+
+This gives a smooth analytical stretching in computational space that is constrained to given specified surface and bottom grid cell thicknesses in real space. This is not to be confused with the hybrid schemes that superimpose geopotential coordinates on terrain following coordinates thus creating a non-analytical vertical coordinate that therefore may suffer from large gradients in the vertical resolutions. This stretching is less straightforward to implement than the \citet{Song_Haidvogel_JCP94} stretching, but has the advantage of resolving diurnal processes in deep water and has generally flatter slopes.
+
+As with the \citet{Song_Haidvogel_JCP94} stretching the stretch is only applied at depths greater than the critical depth $h_c$. In this example two options are available in depths shallower than $h_c$, with pure sigma being applied if the \np{ln\_sigcrit} is true and pure z-coordinates if it is false (the z-coordinate being equal to the depths of the stretched coordinate at $h_c$.
+
+Minimising the horizontal slope of the vertical coordinate is important in terrain-following systems as large slopes lead to hydrostatic consistency. A hydrostatic consistency parameter diagnostic following \citet{Haney1991} has been implemented, and is output as part of the model mesh file at the start of the run.
+
+% -------------------------------------------------------------------------------------------------------------
+% z*- or s*-coordinate
+% -------------------------------------------------------------------------------------------------------------
+\subsection{$z^*$- or $s^*$-coordinate (add \key{vvl}) }
+\label{DOM_zgr_vvl}
+
+This option is described in the Report by Levier \textit{et al.} (2007), available on
+the \NEMO web site.
+
+%gm% key advantage: minimise the diffusion/dispertion associated with advection in response to high frequency surface disturbances
+
+% -------------------------------------------------------------------------------------------------------------
+% level bathymetry and mask
+% -------------------------------------------------------------------------------------------------------------
+\subsection{level bathymetry and mask}
+\label{DOM_msk}
+
+Whatever the vertical coordinate used, the model offers the possibility of
+representing the bottom topography with steps that follow the face of the
+model cells (step like topography) \citep{Madec_al_JPO96}. The distribution of
+the steps in the horizontal is defined in a 2D integer array, mbathy, which
+gives the number of ocean levels ($i.e.$ those that are not masked) at each
+$t$-point. mbathy is computed from the meter bathymetry using the definiton of
+gdept as the number of $t$-points which gdept $\leq$ bathy.
+
+Modifications of the model bathymetry are performed in the \textit{bat\_ctl}
+routine (see \mdl{domzgr} module) after mbathy is computed. Isolated grid points
+that do not communicate with another ocean point at the same level are eliminated.
+
+From the \textit{mbathy} array, the mask fields are defined as follows:
+\begin{align*}
+tmask(i,j,k) &= \begin{cases} \; 1& \text{ if $k\leq mbathy(i,j)$ } \\
+ \; 0& \text{ if $k\leq mbathy(i,j)$ } \end{cases} \\
+umask(i,j,k) &= \; tmask(i,j,k) \ * \ tmask(i+1,j,k) \\
+vmask(i,j,k) &= \; tmask(i,j,k) \ * \ tmask(i,j+1,k) \\
+fmask(i,j,k) &= \; tmask(i,j,k) \ * \ tmask(i+1,j,k) \\
+ & \ \ \, * tmask(i,j,k) \ * \ tmask(i+1,j,k)
+\end{align*}
+
+Note that \textit{wmask} is not defined as it is exactly equal to \textit{tmask} with
+the numerical indexing used (\S~\ref{DOM_Num_Index}). Moreover, the
+specification of closed lateral boundaries requires that at least the first and last
+rows and columns of the \textit{mbathy} array are set to zero. In the particular
+case of an east-west cyclical boundary condition, \textit{mbathy} has its last
+column equal to the second one and its first column equal to the last but one
+(and so too the mask arrays) (see \S~\ref{LBC_jperio}).
+
+%%%
+\gmcomment{ \colorbox{yellow}{Add one word on tricky trick !} mbathy in further modified in zdfbfr{\ldots}. }
+%%%
+
+% ================================================================
+% Domain: Initial State (dtatsd & istate)
+% ================================================================
+\section [Domain: Initial State (\textit{istate and dtatsd})]
+ {Domain: Initial State \small{(\mdl{istate} and \mdl{dtatsd} modules)} }
+\label{DTA_tsd}
+%-----------------------------------------namtsd-------------------------------------------
+\namdisplay{namtsd}
+%------------------------------------------------------------------------------------------
+
+Options are defined in \ngn{namtsd}.
+By default, the ocean start from rest (the velocity field is set to zero) and the initialization of
+temperature and salinity fields is controlled through the \np{ln\_tsd\_ini} namelist parameter.
+\begin{description}
+\item[ln\_tsd\_init = .true.] use a T and S input files that can be given on the model grid itself or
+on their native input data grid. In the latter case, the data will be interpolated on-the-fly both in the
+horizontal and the vertical to the model grid (see \S~\ref{SBC_iof}). The information relative to the
+input files are given in the \np{sn\_tem} and \np{sn\_sal} structures.
+The computation is done in the \mdl{dtatsd} module.
+\item[ln\_tsd\_init = .false.] use constant salinity value of 35.5 psu and an analytical profile of temperature
+(typical of the tropical ocean), see \rou{istate\_t\_s} subroutine called from \mdl{istate} module.
+\end{description}
Index: /branches/UKMO/dev_r5518_GO6_under_ice_relax_dr_hook/DOC/TexFiles/Chapters/Chap_DYN.tex
===================================================================
--- /branches/UKMO/dev_r5518_GO6_under_ice_relax_dr_hook/DOC/TexFiles/Chapters/Chap_DYN.tex (revision 11643)
+++ /branches/UKMO/dev_r5518_GO6_under_ice_relax_dr_hook/DOC/TexFiles/Chapters/Chap_DYN.tex (revision 11643)
@@ -0,0 +1,1297 @@
+% ================================================================
+% Chapter � Ocean Dynamics (DYN)
+% ================================================================
+\chapter{Ocean Dynamics (DYN)}
+\label{DYN}
+\minitoc
+
+% add a figure for dynvor ens, ene latices
+
+%\vspace{2.cm}
+$\ $\newline %force an empty line
+
+Using the representation described in Chapter \ref{DOM}, several semi-discrete
+space forms of the dynamical equations are available depending on the vertical
+coordinate used and on the conservation properties of the vorticity term. In all
+the equations presented here, the masking has been omitted for simplicity.
+One must be aware that all the quantities are masked fields and that each time an
+average or difference operator is used, the resulting field is multiplied by a mask.
+
+The prognostic ocean dynamics equation can be summarized as follows:
+\begin{equation*}
+\text{NXT} = \dbinom {\text{VOR} + \text{KEG} + \text {ZAD} }
+ {\text{COR} + \text{ADV} }
+ + \text{HPG} + \text{SPG} + \text{LDF} + \text{ZDF}
+\end{equation*}
+NXT stands for next, referring to the time-stepping. The first group of terms on
+the rhs of this equation corresponds to the Coriolis and advection
+terms that are decomposed into either a vorticity part (VOR), a kinetic energy part (KEG)
+and a vertical advection part (ZAD) in the vector invariant formulation, or a Coriolis
+and advection part (COR+ADV) in the flux formulation. The terms following these
+are the pressure gradient contributions (HPG, Hydrostatic Pressure Gradient,
+and SPG, Surface Pressure Gradient); and contributions from lateral diffusion
+(LDF) and vertical diffusion (ZDF), which are added to the rhs in the \mdl{dynldf}
+and \mdl{dynzdf} modules. The vertical diffusion term includes the surface and
+bottom stresses. The external forcings and parameterisations require complex
+inputs (surface wind stress calculation using bulk formulae, estimation of mixing
+coefficients) that are carried out in modules SBC, LDF and ZDF and are described
+in Chapters \ref{SBC}, \ref{LDF} and \ref{ZDF}, respectively.
+
+In the present chapter we also describe the diagnostic equations used to compute
+the horizontal divergence, curl of the velocities (\emph{divcur} module) and
+the vertical velocity (\emph{wzvmod} module).
+
+The different options available to the user are managed by namelist variables.
+For term \textit{ttt} in the momentum equations, the logical namelist variables are \textit{ln\_dynttt\_xxx},
+where \textit{xxx} is a 3 or 4 letter acronym corresponding to each optional scheme.
+If a CPP key is used for this term its name is \textbf{key\_ttt}. The corresponding
+code can be found in the \textit{dynttt\_xxx} module in the DYN directory, and it is
+usually computed in the \textit{dyn\_ttt\_xxx} subroutine.
+
+The user has the option of extracting and outputting each tendency term from the
+3D momentum equations (\key{trddyn} defined), as described in
+Chap.\ref{MISC}. Furthermore, the tendency terms associated with the 2D
+barotropic vorticity balance (when \key{trdvor} is defined) can be derived from the
+3D terms.
+%%%
+\gmcomment{STEVEN: not quite sure I've got the sense of the last sentence. does
+MISC correspond to "extracting tendency terms" or "vorticity balance"?}
+
+$\ $\newline % force a new ligne
+
+% ================================================================
+% Sea Surface Height evolution & Diagnostics variables
+% ================================================================
+\section{Sea surface height and diagnostic variables ($\eta$, $\zeta$, $\chi$, $w$)}
+\label{DYN_divcur_wzv}
+
+%--------------------------------------------------------------------------------------------------------------
+% Horizontal divergence and relative vorticity
+%--------------------------------------------------------------------------------------------------------------
+\subsection [Horizontal divergence and relative vorticity (\textit{divcur})]
+ {Horizontal divergence and relative vorticity (\mdl{divcur})}
+\label{DYN_divcur}
+
+The vorticity is defined at an $f$-point ($i.e.$ corner point) as follows:
+\begin{equation} \label{Eq_divcur_cur}
+\zeta =\frac{1}{e_{1f}\,e_{2f} }\left( {\;\delta _{i+1/2} \left[ {e_{2v}\;v} \right]
+ -\delta _{j+1/2} \left[ {e_{1u}\;u} \right]\;} \right)
+\end{equation}
+
+The horizontal divergence is defined at a $T$-point. It is given by:
+\begin{equation} \label{Eq_divcur_div}
+\chi =\frac{1}{e_{1t}\,e_{2t}\,e_{3t} }
+ \left( {\delta _i \left[ {e_{2u}\,e_{3u}\,u} \right]
+ +\delta _j \left[ {e_{1v}\,e_{3v}\,v} \right]} \right)
+\end{equation}
+
+Note that although the vorticity has the same discrete expression in $z$-
+and $s$-coordinates, its physical meaning is not identical. $\zeta$ is a pseudo
+vorticity along $s$-surfaces (only pseudo because $(u,v)$ are still defined along
+geopotential surfaces, but are not necessarily defined at the same depth).
+
+The vorticity and divergence at the \textit{before} step are used in the computation
+of the horizontal diffusion of momentum. Note that because they have been
+calculated prior to the Asselin filtering of the \textit{before} velocities, the
+\textit{before} vorticity and divergence arrays must be included in the restart file
+to ensure perfect restartability. The vorticity and divergence at the \textit{now}
+time step are used for the computation of the nonlinear advection and of the
+vertical velocity respectively.
+
+%--------------------------------------------------------------------------------------------------------------
+% Sea Surface Height evolution
+%--------------------------------------------------------------------------------------------------------------
+\subsection [Sea surface height evolution and vertical velocity (\textit{sshwzv})]
+ {Horizontal divergence and relative vorticity (\mdl{sshwzv})}
+\label{DYN_sshwzv}
+
+The sea surface height is given by :
+\begin{equation} \label{Eq_dynspg_ssh}
+\begin{aligned}
+\frac{\partial \eta }{\partial t}
+&\equiv \frac{1}{e_{1t} e_{2t} }\sum\limits_k { \left\{ \delta _i \left[ {e_{2u}\,e_{3u}\;u} \right]
+ +\delta _j \left[ {e_{1v}\,e_{3v}\;v} \right] \right\} }
+ - \frac{\textit{emp}}{\rho _w } \\
+&\equiv \sum\limits_k {\chi \ e_{3t}} - \frac{\textit{emp}}{\rho _w }
+\end{aligned}
+\end{equation}
+where \textit{emp} is the surface freshwater budget (evaporation minus precipitation),
+expressed in Kg/m$^2$/s (which is equal to mm/s), and $\rho _w$=1,035~Kg/m$^3$
+is the reference density of sea water (Boussinesq approximation). If river runoff is
+expressed as a surface freshwater flux (see \S\ref{SBC}) then \textit{emp} can be
+written as the evaporation minus precipitation, minus the river runoff.
+The sea-surface height is evaluated using exactly the same time stepping scheme
+as the tracer equation \eqref{Eq_tra_nxt}:
+a leapfrog scheme in combination with an Asselin time filter, $i.e.$ the velocity appearing
+in \eqref{Eq_dynspg_ssh} is centred in time (\textit{now} velocity).
+This is of paramount importance. Replacing $T$ by the number $1$ in the tracer equation and summing
+over the water column must lead to the sea surface height equation otherwise tracer content
+will not be conserved \citep{Griffies_al_MWR01, Leclair_Madec_OM09}.
+
+The vertical velocity is computed by an upward integration of the horizontal
+divergence starting at the bottom, taking into account the change of the thickness of the levels :
+\begin{equation} \label{Eq_wzv}
+\left\{ \begin{aligned}
+&\left. w \right|_{k_b-1/2} \quad= 0 \qquad \text{where } k_b \text{ is the level just above the sea floor } \\
+&\left. w \right|_{k+1/2} = \left. w \right|_{k-1/2} + \left. e_{3t} \right|_{k}\; \left. \chi \right|_k
+ - \frac{1} {2 \rdt} \left( \left. e_{3t}^{t+1}\right|_{k} - \left. e_{3t}^{t-1}\right|_{k}\right)
+\end{aligned} \right.
+\end{equation}
+
+In the case of a non-linear free surface (\key{vvl}), the top vertical velocity is $-\textit{emp}/\rho_w$,
+as changes in the divergence of the barotropic transport are absorbed into the change
+of the level thicknesses, re-orientated downward.
+\gmcomment{not sure of this... to be modified with the change in emp setting}
+In the case of a linear free surface, the time derivative in \eqref{Eq_wzv} disappears.
+The upper boundary condition applies at a fixed level $z=0$. The top vertical velocity
+is thus equal to the divergence of the barotropic transport ($i.e.$ the first term in the
+right-hand-side of \eqref{Eq_dynspg_ssh}).
+
+Note also that whereas the vertical velocity has the same discrete
+expression in $z$- and $s$-coordinates, its physical meaning is not the same:
+in the second case, $w$ is the velocity normal to the $s$-surfaces.
+Note also that the $k$-axis is re-orientated downwards in the \textsc{fortran} code compared
+to the indexing used in the semi-discrete equations such as \eqref{Eq_wzv}
+(see \S\ref{DOM_Num_Index_vertical}).
+
+
+% ================================================================
+% Coriolis and Advection terms: vector invariant form
+% ================================================================
+\section{Coriolis and Advection: vector invariant form}
+\label{DYN_adv_cor_vect}
+%-----------------------------------------nam_dynadv----------------------------------------------------
+\namdisplay{namdyn_adv}
+%-------------------------------------------------------------------------------------------------------------
+
+The vector invariant form of the momentum equations is the one most
+often used in applications of the \NEMO ocean model. The flux form option
+(see next section) has been present since version $2$. Options are defined
+through the \ngn{namdyn\_adv} namelist variables
+Coriolis and momentum advection terms are evaluated using a leapfrog
+scheme, $i.e.$ the velocity appearing in these expressions is centred in
+time (\textit{now} velocity).
+At the lateral boundaries either free slip, no slip or partial slip boundary
+conditions are applied following Chap.\ref{LBC}.
+
+% -------------------------------------------------------------------------------------------------------------
+% Vorticity term
+% -------------------------------------------------------------------------------------------------------------
+\subsection [Vorticity term (\textit{dynvor}) ]
+ {Vorticity term (\mdl{dynvor})}
+\label{DYN_vor}
+%------------------------------------------nam_dynvor----------------------------------------------------
+\namdisplay{namdyn_vor}
+%-------------------------------------------------------------------------------------------------------------
+
+Options are defined through the \ngn{namdyn\_vor} namelist variables.
+Four discretisations of the vorticity term (\textit{ln\_dynvor\_xxx}=true) are available:
+conserving potential enstrophy of horizontally non-divergent flow (ENS scheme) ;
+conserving horizontal kinetic energy (ENE scheme) ; conserving potential enstrophy for
+the relative vorticity term and horizontal kinetic energy for the planetary vorticity
+term (MIX scheme) ; or conserving both the potential enstrophy of horizontally non-divergent
+flow and horizontal kinetic energy (EEN scheme) (see Appendix~\ref{Apdx_C_vorEEN}). In the
+case of ENS, ENE or MIX schemes the land sea mask may be slightly modified to ensure the
+consistency of vorticity term with analytical equations (\textit{ln\_dynvor\_con}=true).
+The vorticity terms are all computed in dedicated routines that can be found in
+the \mdl{dynvor} module.
+
+%-------------------------------------------------------------
+% enstrophy conserving scheme
+%-------------------------------------------------------------
+\subsubsection{Enstrophy conserving scheme (\np{ln\_dynvor\_ens}=true)}
+\label{DYN_vor_ens}
+
+In the enstrophy conserving case (ENS scheme), the discrete formulation of the
+vorticity term provides a global conservation of the enstrophy
+($ [ (\zeta +f ) / e_{3f} ]^2 $ in $s$-coordinates) for a horizontally non-divergent
+flow ($i.e.$ $\chi$=$0$), but does not conserve the total kinetic energy. It is given by:
+\begin{equation} \label{Eq_dynvor_ens}
+\left\{
+\begin{aligned}
+{+\frac{1}{e_{1u} } } & {\overline {\left( { \frac{\zeta +f}{e_{3f} }} \right)} }^{\,i}
+ & {\overline{\overline {\left( {e_{1v}\,e_{3v}\;v} \right)}} }^{\,i, j+1/2} \\
+{- \frac{1}{e_{2v} } } & {\overline {\left( {\frac{\zeta +f}{e_{3f} }} \right)} }^{\,j}
+ & {\overline{\overline {\left( {e_{2u}\,e_{3u}\;u} \right)}} }^{\,i+1/2, j}
+\end{aligned}
+ \right.
+\end{equation}
+
+%-------------------------------------------------------------
+% energy conserving scheme
+%-------------------------------------------------------------
+\subsubsection{Energy conserving scheme (\np{ln\_dynvor\_ene}=true)}
+\label{DYN_vor_ene}
+
+The kinetic energy conserving scheme (ENE scheme) conserves the global
+kinetic energy but not the global enstrophy. It is given by:
+\begin{equation} \label{Eq_dynvor_ene}
+\left\{ \begin{aligned}
+{+\frac{1}{e_{1u}}\; {\overline {\left( {\frac{\zeta +f}{e_{3f} }} \right)
+ \; \overline {\left( {e_{1v}\,e_{3v}\;v} \right)} ^{\,i+1/2}} }^{\,j} } \\
+{- \frac{1}{e_{2v}}\; {\overline {\left( {\frac{\zeta +f}{e_{3f} }} \right)
+ \; \overline {\left( {e_{2u}\,e_{3u}\;u} \right)} ^{\,j+1/2}} }^{\,i} }
+\end{aligned} \right.
+\end{equation}
+
+%-------------------------------------------------------------
+% mix energy/enstrophy conserving scheme
+%-------------------------------------------------------------
+\subsubsection{Mixed energy/enstrophy conserving scheme (\np{ln\_dynvor\_mix}=true) }
+\label{DYN_vor_mix}
+
+For the mixed energy/enstrophy conserving scheme (MIX scheme), a mixture of the
+two previous schemes is used. It consists of the ENS scheme (\ref{Eq_dynvor_ens})
+for the relative vorticity term, and of the ENE scheme (\ref{Eq_dynvor_ene}) applied
+to the planetary vorticity term.
+\begin{equation} \label{Eq_dynvor_mix}
+\left\{ { \begin{aligned}
+ {+\frac{1}{e_{1u} }\; {\overline {\left( {\frac{\zeta }{e_{3f} }} \right)} }^{\,i}
+ \; {\overline{\overline {\left( {e_{1v}\,e_{3v}\;v} \right)}} }^{\,i,j+1/2} -\frac{1}{e_{1u} }
+ \; {\overline {\left( {\frac{f}{e_{3f} }} \right)
+ \;\overline {\left( {e_{1v}\,e_{3v}\;v} \right)} ^{\,i+1/2}} }^{\,j} } \\
+{-\frac{1}{e_{2v} }\; {\overline {\left( {\frac{\zeta }{e_{3f} }} \right)} }^j
+ \; {\overline{\overline {\left( {e_{2u}\,e_{3u}\;u} \right)}} }^{\,i+1/2,j} +\frac{1}{e_{2v} }
+ \; {\overline {\left( {\frac{f}{e_{3f} }} \right)
+ \;\overline {\left( {e_{2u}\,e_{3u}\;u} \right)} ^{\,j+1/2}} }^{\,i} } \hfill
+\end{aligned} } \right.
+\end{equation}
+
+%-------------------------------------------------------------
+% energy and enstrophy conserving scheme
+%-------------------------------------------------------------
+\subsubsection{Energy and enstrophy conserving scheme (\np{ln\_dynvor\_een}=true) }
+\label{DYN_vor_een}
+
+In both the ENS and ENE schemes, it is apparent that the combination of $i$ and $j$
+averages of the velocity allows for the presence of grid point oscillation structures
+that will be invisible to the operator. These structures are \textit{computational modes}
+that will be at least partly damped by the momentum diffusion operator ($i.e.$ the
+subgrid-scale advection), but not by the resolved advection term. The ENS and ENE schemes
+therefore do not contribute to dump any grid point noise in the horizontal velocity field.
+Such noise would result in more noise in the vertical velocity field, an undesirable feature.
+This is a well-known characteristic of $C$-grid discretization where $u$ and $v$ are located
+at different grid points, a price worth paying to avoid a double averaging in the pressure
+gradient term as in the $B$-grid.
+\gmcomment{ To circumvent this, Adcroft (ADD REF HERE)
+Nevertheless, this technique strongly distort the phase and group velocity of Rossby waves....}
+
+A very nice solution to the problem of double averaging was proposed by \citet{Arakawa_Hsu_MWR90}.
+The idea is to get rid of the double averaging by considering triad combinations of vorticity.
+It is noteworthy that this solution is conceptually quite similar to the one proposed by
+\citep{Griffies_al_JPO98} for the discretization of the iso-neutral diffusion operator (see App.\ref{Apdx_C}).
+
+The \citet{Arakawa_Hsu_MWR90} vorticity advection scheme for a single layer is modified
+for spherical coordinates as described by \citet{Arakawa_Lamb_MWR81} to obtain the EEN scheme.
+First consider the discrete expression of the potential vorticity, $q$, defined at an $f$-point:
+\begin{equation} \label{Eq_pot_vor}
+q = \frac{\zeta +f} {e_{3f} }
+\end{equation}
+where the relative vorticity is defined by (\ref{Eq_divcur_cur}), the Coriolis parameter
+is given by $f=2 \,\Omega \;\sin \varphi _f $ and the layer thickness at $f$-points is:
+\begin{equation} \label{Eq_een_e3f}
+e_{3f} = \overline{\overline {e_{3t} }} ^{\,i+1/2,j+1/2}
+\end{equation}
+
+%>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>
+\begin{figure}[!ht] \begin{center}
+\includegraphics[width=0.70\textwidth]{./TexFiles/Figures/Fig_DYN_een_triad.pdf}
+\caption{ \label{Fig_DYN_een_triad}
+Triads used in the energy and enstrophy conserving scheme (een) for
+$u$-component (upper panel) and $v$-component (lower panel).}
+\end{center} \end{figure}
+%>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>
+
+Note that a key point in \eqref{Eq_een_e3f} is that the averaging in the \textbf{i}- and
+\textbf{j}- directions uses the masked vertical scale factor but is always divided by
+$4$, not by the sum of the masks at the four $T$-points. This preserves the continuity of
+$e_{3f}$ when one or more of the neighbouring $e_{3t}$ tends to zero and
+extends by continuity the value of $e_{3f}$ into the land areas. This feature is essential for
+the $z$-coordinate with partial steps.
+
+Next, the vorticity triads, $ {^i_j}\mathbb{Q}^{i_p}_{j_p}$ can be defined at a $T$-point as
+the following triad combinations of the neighbouring potential vorticities defined at f-points
+(Fig.~\ref{Fig_DYN_een_triad}):
+\begin{equation} \label{Q_triads}
+_i^j \mathbb{Q}^{i_p}_{j_p}
+= \frac{1}{12} \ \left( q^{i-i_p}_{j+j_p} + q^{i+j_p}_{j+i_p} + q^{i+i_p}_{j-j_p} \right)
+\end{equation}
+where the indices $i_p$ and $k_p$ take the values: $i_p = -1/2$ or $1/2$ and $j_p = -1/2$ or $1/2$.
+
+Finally, the vorticity terms are represented as:
+\begin{equation} \label{Eq_dynvor_een}
+\left\{ {
+\begin{aligned}
+ +q\,e_3 \, v &\equiv +\frac{1}{e_{1u} } \sum_{\substack{i_p,\,k_p}}
+ {^{i+1/2-i_p}_j} \mathbb{Q}^{i_p}_{j_p} \left( e_{1v}\,e_{3v} \;v \right)^{i+1/2-i_p}_{j+j_p} \\
+ - q\,e_3 \, u &\equiv -\frac{1}{e_{2v} } \sum_{\substack{i_p,\,k_p}}
+ {^i_{j+1/2-j_p}} \mathbb{Q}^{i_p}_{j_p} \left( e_{2u}\,e_{3u} \;u \right)^{i+i_p}_{j+1/2-j_p} \\
+\end{aligned}
+} \right.
+\end{equation}
+
+This EEN scheme in fact combines the conservation properties of the ENS and ENE schemes.
+It conserves both total energy and potential enstrophy in the limit of horizontally
+nondivergent flow ($i.e.$ $\chi$=$0$) (see Appendix~\ref{Apdx_C_vorEEN}).
+Applied to a realistic ocean configuration, it has been shown that it leads to a significant
+reduction of the noise in the vertical velocity field \citep{Le_Sommer_al_OM09}.
+Furthermore, used in combination with a partial steps representation of bottom topography,
+it improves the interaction between current and topography, leading to a larger
+topostrophy of the flow \citep{Barnier_al_OD06, Penduff_al_OS07}.
+
+%--------------------------------------------------------------------------------------------------------------
+% Kinetic Energy Gradient term
+%--------------------------------------------------------------------------------------------------------------
+\subsection [Kinetic Energy Gradient term (\textit{dynkeg})]
+ {Kinetic Energy Gradient term (\mdl{dynkeg})}
+\label{DYN_keg}
+
+As demonstrated in Appendix~\ref{Apdx_C}, there is a single discrete formulation
+of the kinetic energy gradient term that, together with the formulation chosen for
+the vertical advection (see below), conserves the total kinetic energy:
+\begin{equation} \label{Eq_dynkeg}
+\left\{ \begin{aligned}
+ -\frac{1}{2 \; e_{1u} } & \ \delta _{i+1/2} \left[ {\overline {u^2}^{\,i} + \overline{v^2}^{\,j}} \right] \\
+ -\frac{1}{2 \; e_{2v} } & \ \delta _{j+1/2} \left[ {\overline {u^2}^{\,i} + \overline{v^2}^{\,j}} \right]
+\end{aligned} \right.
+\end{equation}
+
+%--------------------------------------------------------------------------------------------------------------
+% Vertical advection term
+%--------------------------------------------------------------------------------------------------------------
+\subsection [Vertical advection term (\textit{dynzad}) ]
+ {Vertical advection term (\mdl{dynzad}) }
+\label{DYN_zad}
+
+The discrete formulation of the vertical advection, together with the formulation
+chosen for the gradient of kinetic energy (KE) term, conserves the total kinetic
+energy. Indeed, the change of KE due to the vertical advection is exactly
+balanced by the change of KE due to the gradient of KE (see Appendix~\ref{Apdx_C}).
+\begin{equation} \label{Eq_dynzad}
+\left\{ \begin{aligned}
+-\frac{1} {e_{1u}\,e_{2u}\,e_{3u}} &\ \overline{\ \overline{ e_{1t}\,e_{2t}\;w } ^{\,i+1/2} \;\delta _{k+1/2} \left[ u \right]\ }^{\,k} \\
+-\frac{1} {e_{1v}\,e_{2v}\,e_{3v}} &\ \overline{\ \overline{ e_{1t}\,e_{2t}\;w } ^{\,j+1/2} \;\delta _{k+1/2} \left[ u \right]\ }^{\,k}
+\end{aligned} \right.
+\end{equation}
+
+% ================================================================
+% Coriolis and Advection : flux form
+% ================================================================
+\section{Coriolis and Advection: flux form}
+\label{DYN_adv_cor_flux}
+%------------------------------------------nam_dynadv----------------------------------------------------
+\namdisplay{namdyn_adv}
+%-------------------------------------------------------------------------------------------------------------
+
+Options are defined through the \ngn{namdyn\_adv} namelist variables.
+In the flux form (as in the vector invariant form), the Coriolis and momentum
+advection terms are evaluated using a leapfrog scheme, $i.e.$ the velocity
+appearing in their expressions is centred in time (\textit{now} velocity). At the
+lateral boundaries either free slip, no slip or partial slip boundary conditions
+are applied following Chap.\ref{LBC}.
+
+
+%--------------------------------------------------------------------------------------------------------------
+% Coriolis plus curvature metric terms
+%--------------------------------------------------------------------------------------------------------------
+\subsection [Coriolis plus curvature metric terms (\textit{dynvor}) ]
+ {Coriolis plus curvature metric terms (\mdl{dynvor}) }
+\label{DYN_cor_flux}
+
+In flux form, the vorticity term reduces to a Coriolis term in which the Coriolis
+parameter has been modified to account for the "metric" term. This altered
+Coriolis parameter is thus discretised at $f$-points. It is given by:
+\begin{multline} \label{Eq_dyncor_metric}
+f+\frac{1}{e_1 e_2 }\left( {v\frac{\partial e_2 }{\partial i} - u\frac{\partial e_1 }{\partial j}} \right) \\
+ \equiv f + \frac{1}{e_{1f} e_{2f} } \left( { \ \overline v ^{i+1/2}\delta _{i+1/2} \left[ {e_{2u} } \right]
+ - \overline u ^{j+1/2}\delta _{j+1/2} \left[ {e_{1u} } \right] } \ \right)
+\end{multline}
+
+Any of the (\ref{Eq_dynvor_ens}), (\ref{Eq_dynvor_ene}) and (\ref{Eq_dynvor_een})
+schemes can be used to compute the product of the Coriolis parameter and the
+vorticity. However, the energy-conserving scheme (\ref{Eq_dynvor_een}) has
+exclusively been used to date. This term is evaluated using a leapfrog scheme,
+$i.e.$ the velocity is centred in time (\textit{now} velocity).
+
+%--------------------------------------------------------------------------------------------------------------
+% Flux form Advection term
+%--------------------------------------------------------------------------------------------------------------
+\subsection [Flux form Advection term (\textit{dynadv}) ]
+ {Flux form Advection term (\mdl{dynadv}) }
+\label{DYN_adv_flux}
+
+The discrete expression of the advection term is given by :
+\begin{equation} \label{Eq_dynadv}
+\left\{
+\begin{aligned}
+\frac{1}{e_{1u}\,e_{2u}\,e_{3u}}
+\left( \delta _{i+1/2} \left[ \overline{e_{2u}\,e_{3u}\;u }^{i } \ u_t \right]
+ + \delta _{j } \left[ \overline{e_{1u}\,e_{3u}\;v }^{i+1/2} \ u_f \right] \right. \ \; \\
+\left. + \delta _{k } \left[ \overline{e_{1w}\,e_{2w}\;w}^{i+1/2} \ u_{uw} \right] \right) \\
+\\
+\frac{1}{e_{1v}\,e_{2v}\,e_{3v}}
+\left( \delta _{i } \left[ \overline{e_{2u}\,e_{3u }\;u }^{j+1/2} \ v_f \right]
+ + \delta _{j+1/2} \left[ \overline{e_{1u}\,e_{3u }\;v }^{i } \ v_t \right] \right. \ \, \, \\
+\left. + \delta _{k } \left[ \overline{e_{1w}\,e_{2w}\;w}^{j+1/2} \ v_{vw} \right] \right) \\
+\end{aligned}
+\right.
+\end{equation}
+
+Two advection schemes are available: a $2^{nd}$ order centered finite
+difference scheme, CEN2, or a $3^{rd}$ order upstream biased scheme, UBS.
+The latter is described in \citet{Shchepetkin_McWilliams_OM05}. The schemes are
+selected using the namelist logicals \np{ln\_dynadv\_cen2} and \np{ln\_dynadv\_ubs}.
+In flux form, the schemes differ by the choice of a space and time interpolation to
+define the value of $u$ and $v$ at the centre of each face of $u$- and $v$-cells,
+$i.e.$ at the $T$-, $f$-, and $uw$-points for $u$ and at the $f$-, $T$- and
+$vw$-points for $v$.
+
+%-------------------------------------------------------------
+% 2nd order centred scheme
+%-------------------------------------------------------------
+\subsubsection{$2^{nd}$ order centred scheme (cen2) (\np{ln\_dynadv\_cen2}=true)}
+\label{DYN_adv_cen2}
+
+In the centered $2^{nd}$ order formulation, the velocity is evaluated as the
+mean of the two neighbouring points :
+\begin{equation} \label{Eq_dynadv_cen2}
+\left\{ \begin{aligned}
+ u_T^{cen2} &=\overline u^{i } \quad & u_F^{cen2} &=\overline u^{j+1/2} \quad & u_{uw}^{cen2} &=\overline u^{k+1/2} \\
+ v_F^{cen2} &=\overline v ^{i+1/2} \quad & v_F^{cen2} &=\overline v^j \quad & v_{vw}^{cen2} &=\overline v ^{k+1/2} \\
+\end{aligned} \right.
+\end{equation}
+
+The scheme is non diffusive (i.e. conserves the kinetic energy) but dispersive
+($i.e.$ it may create false extrema). It is therefore notoriously noisy and must be
+used in conjunction with an explicit diffusion operator to produce a sensible solution.
+The associated time-stepping is performed using a leapfrog scheme in conjunction
+with an Asselin time-filter, so $u$ and $v$ are the \emph{now} velocities.
+
+%-------------------------------------------------------------
+% UBS scheme
+%-------------------------------------------------------------
+\subsubsection{Upstream Biased Scheme (UBS) (\np{ln\_dynadv\_ubs}=true)}
+\label{DYN_adv_ubs}
+
+The UBS advection scheme is an upstream biased third order scheme based on
+an upstream-biased parabolic interpolation. For example, the evaluation of
+$u_T^{ubs} $ is done as follows:
+\begin{equation} \label{Eq_dynadv_ubs}
+u_T^{ubs} =\overline u ^i-\;\frac{1}{6} \begin{cases}
+ u"_{i-1/2}& \text{if $\ \overline{e_{2u}\,e_{3u} \ u}^i \geqslant 0$ } \\
+ u"_{i+1/2}& \text{if $\ \overline{e_{2u}\,e_{3u} \ u}^i < 0$ }
+\end{cases}
+\end{equation}
+where $u"_{i+1/2} =\delta _{i+1/2} \left[ {\delta _i \left[ u \right]} \right]$. This results
+in a dissipatively dominant ($i.e.$ hyper-diffusive) truncation error \citep{Shchepetkin_McWilliams_OM05}.
+The overall performance of the advection scheme is similar to that reported in
+\citet{Farrow1995}. It is a relatively good compromise between accuracy and
+smoothness. It is not a \emph{positive} scheme, meaning that false extrema are
+permitted. But the amplitudes of the false extrema are significantly reduced over
+those in the centred second order method. As the scheme already includes
+a diffusion component, it can be used without explicit lateral diffusion on momentum
+($i.e.$ \np{ln\_dynldf\_lap}=\np{ln\_dynldf\_bilap}=false), and it is recommended to do so.
+
+The UBS scheme is not used in all directions. In the vertical, the centred $2^{nd}$
+order evaluation of the advection is preferred, $i.e.$ $u_{uw}^{ubs}$ and
+$u_{vw}^{ubs}$ in \eqref{Eq_dynadv_cen2} are used. UBS is diffusive and is
+associated with vertical mixing of momentum. \gmcomment{ gm pursue the
+sentence:Since vertical mixing of momentum is a source term of the TKE equation... }
+
+For stability reasons, the first term in (\ref{Eq_dynadv_ubs}), which corresponds
+to a second order centred scheme, is evaluated using the \textit{now} velocity
+(centred in time), while the second term, which is the diffusion part of the scheme,
+is evaluated using the \textit{before} velocity (forward in time). This is discussed
+by \citet{Webb_al_JAOT98} in the context of the Quick advection scheme.
+
+Note that the UBS and QUICK (Quadratic Upstream Interpolation for Convective Kinematics)
+schemes only differ by one coefficient. Replacing $1/6$ by $1/8$ in
+(\ref{Eq_dynadv_ubs}) leads to the QUICK advection scheme \citep{Webb_al_JAOT98}.
+This option is not available through a namelist parameter, since the $1/6$ coefficient
+is hard coded. Nevertheless it is quite easy to make the substitution in the
+\mdl{dynadv\_ubs} module and obtain a QUICK scheme.
+
+Note also that in the current version of \mdl{dynadv\_ubs}, there is also the
+possibility of using a $4^{th}$ order evaluation of the advective velocity as in
+ROMS. This is an error and should be suppressed soon.
+%%%
+\gmcomment{action : this have to be done}
+%%%
+
+% ================================================================
+% Hydrostatic pressure gradient term
+% ================================================================
+\section [Hydrostatic pressure gradient (\textit{dynhpg})]
+ {Hydrostatic pressure gradient (\mdl{dynhpg})}
+\label{DYN_hpg}
+%------------------------------------------nam_dynhpg---------------------------------------------------
+\namdisplay{namdyn_hpg}
+%-------------------------------------------------------------------------------------------------------------
+
+Options are defined through the \ngn{namdyn\_hpg} namelist variables.
+The key distinction between the different algorithms used for the hydrostatic
+pressure gradient is the vertical coordinate used, since HPG is a \emph{horizontal}
+pressure gradient, $i.e.$ computed along geopotential surfaces. As a result, any
+tilt of the surface of the computational levels will require a specific treatment to
+compute the hydrostatic pressure gradient.
+
+The hydrostatic pressure gradient term is evaluated either using a leapfrog scheme,
+$i.e.$ the density appearing in its expression is centred in time (\emph{now} $\rho$), or
+a semi-implcit scheme. At the lateral boundaries either free slip, no slip or partial slip
+boundary conditions are applied.
+
+%--------------------------------------------------------------------------------------------------------------
+% z-coordinate with full step
+%--------------------------------------------------------------------------------------------------------------
+\subsection [$z$-coordinate with full step (\np{ln\_dynhpg\_zco}) ]
+ {$z$-coordinate with full step (\np{ln\_dynhpg\_zco}=true)}
+\label{DYN_hpg_zco}
+
+The hydrostatic pressure can be obtained by integrating the hydrostatic equation
+vertically from the surface. However, the pressure is large at great depth while its
+horizontal gradient is several orders of magnitude smaller. This may lead to large
+truncation errors in the pressure gradient terms. Thus, the two horizontal components
+of the hydrostatic pressure gradient are computed directly as follows:
+
+for $k=km$ (surface layer, $jk=1$ in the code)
+\begin{equation} \label{Eq_dynhpg_zco_surf}
+\left\{ \begin{aligned}
+ \left. \delta _{i+1/2} \left[ p^h \right] \right|_{k=km}
+&= \frac{1}{2} g \ \left. \delta _{i+1/2} \left[ e_{3w} \ \rho \right] \right|_{k=km} \\
+ \left. \delta _{j+1/2} \left[ p^h \right] \right|_{k=km}
+&= \frac{1}{2} g \ \left. \delta _{j+1/2} \left[ e_{3w} \ \rho \right] \right|_{k=km} \\
+\end{aligned} \right.
+\end{equation}
+
+for $1 > > > > > > > > > > > > > > > > > > > > > > > > > > >
+\begin{figure}[!t] \begin{center}
+\includegraphics[width=0.7\textwidth]{./TexFiles/Figures/Fig_DYN_dynspg_ts.pdf}
+\caption{ \label{Fig_DYN_dynspg_ts}
+Schematic of the split-explicit time stepping scheme for the external
+and internal modes. Time increases to the right. In this particular exemple,
+a boxcar averaging window over $nn\_baro$ barotropic time steps is used ($nn\_bt\_filt=1$) and $nn\_baro=5$.
+Internal mode time steps (which are also the model time steps) are denoted
+by $t-\rdt$, $t$ and $t+\rdt$. Variables with $k$ superscript refer to instantaneous barotropic variables,
+$< >$ and $<< >>$ operator refer to time filtered variables using respectively primary (red vertical bars) and secondary weights (blue vertical bars).
+The former are used to obtain time filtered quantities at $t+\rdt$ while the latter are used to obtain time averaged
+transports to advect tracers.
+a) Forward time integration: \np{ln\_bt\_fw}=true, \np{ln\_bt\_ave}=true.
+b) Centred time integration: \np{ln\_bt\_fw}=false, \np{ln\_bt\_ave}=true.
+c) Forward time integration with no time filtering (POM-like scheme): \np{ln\_bt\_fw}=true, \np{ln\_bt\_ave}=false. }
+\end{center} \end{figure}
+%> > > > > > > > > > > > > > > > > > > > > > > > > > > >
+
+In the default case (\np{ln\_bt\_fw}=true), the external mode is integrated
+between \textit{now} and \textit{after} baroclinic time-steps (Fig.~\ref{Fig_DYN_dynspg_ts}a). To avoid aliasing of fast barotropic motions into three dimensional equations, time filtering is eventually applied on barotropic
+quantities (\np{ln\_bt\_ave}=true). In that case, the integration is extended slightly beyond \textit{after} time step to provide time filtered quantities.
+These are used for the subsequent initialization of the barotropic mode in the following baroclinic step.
+Since external mode equations written at baroclinic time steps finally follow a forward time stepping scheme,
+asselin filtering is not applied to barotropic quantities. \\
+Alternatively, one can choose to integrate barotropic equations starting
+from \textit{before} time step (\np{ln\_bt\_fw}=false). Although more computationaly expensive ( \np{nn\_baro} additional iterations are indeed necessary), the baroclinic to barotropic forcing term given at \textit{now} time step
+become centred in the middle of the integration window. It can easily be shown that this property
+removes part of splitting errors between modes, which increases the overall numerical robustness.
+%references to Patrick Marsaleix' work here. Also work done by SHOM group.
+
+%%%
+
+As far as tracer conservation is concerned, barotropic velocities used to advect tracers must also be updated
+at \textit{now} time step. This implies to change the traditional order of computations in \NEMO: most of momentum
+trends (including the barotropic mode calculation) updated first, tracers' after. This \textit{de facto} makes semi-implicit hydrostatic
+pressure gradient (see section \S\ref{DYN_hpg_imp}) and time splitting not compatible.
+Advective barotropic velocities are obtained by using a secondary set of filtering weights, uniquely defined from the filter
+coefficients used for the time averaging (\citet{Shchepetkin_McWilliams_OM05}). Consistency between the time averaged continuity equation and the time stepping of tracers is here the key to obtain exact conservation.
+
+%%%
+
+One can eventually choose to feedback instantaneous values by not using any time filter (\np{ln\_bt\_ave}=false).
+In that case, external mode equations are continuous in time, ie they are not re-initialized when starting a new
+sub-stepping sequence. This is the method used so far in the POM model, the stability being maintained by refreshing at (almost)
+each barotropic time step advection and horizontal diffusion terms. Since the latter terms have not been added in \NEMO for
+computational efficiency, removing time filtering is not recommended except for debugging purposes.
+This may be used for instance to appreciate the damping effect of the standard formulation on external gravity waves in idealized or weakly non-linear cases. Although the damping is lower than for the filtered free surface, it is still significant as shown by \citet{Levier2007} in the case of an analytical barotropic Kelvin wave.
+
+%>>>>>===============
+\gmcomment{ %%% copy from griffies Book
+
+\textbf{title: Time stepping the barotropic system }
+
+Assume knowledge of the full velocity and tracer fields at baroclinic time $\tau$. Hence,
+we can update the surface height and vertically integrated velocity with a leap-frog
+scheme using the small barotropic time step $\rdt$. We have
+
+\begin{equation} \label{DYN_spg_ts_eta}
+\eta^{(b)}(\tau,t_{n+1}) - \eta^{(b)}(\tau,t_{n+1}) (\tau,t_{n-1})
+ = 2 \rdt \left[-\nabla \cdot \textbf{U}^{(b)}(\tau,t_n) + \text{EMP}_w(\tau) \right]
+\end{equation}
+\begin{multline} \label{DYN_spg_ts_u}
+\textbf{U}^{(b)}(\tau,t_{n+1}) - \textbf{U}^{(b)}(\tau,t_{n-1}) \\
+ = 2\rdt \left[ - f \textbf{k} \times \textbf{U}^{(b)}(\tau,t_{n})
+ - H(\tau) \nabla p_s^{(b)}(\tau,t_{n}) +\textbf{M}(\tau) \right]
+\end{multline}
+\
+
+In these equations, araised (b) denotes values of surface height and vertically integrated velocity updated with the barotropic time steps. The $\tau$ time label on $\eta^{(b)}$
+and $U^{(b)}$ denotes the baroclinic time at which the vertically integrated forcing $\textbf{M}(\tau)$ (note that this forcing includes the surface freshwater forcing), the tracer fields, the freshwater flux $\text{EMP}_w(\tau)$, and total depth of the ocean $H(\tau)$ are held for the duration of the barotropic time stepping over a single cycle. This is also the time
+that sets the barotropic time steps via
+\begin{equation} \label{DYN_spg_ts_t}
+t_n=\tau+n\rdt
+\end{equation}
+with $n$ an integer. The density scaled surface pressure is evaluated via
+\begin{equation} \label{DYN_spg_ts_ps}
+p_s^{(b)}(\tau,t_{n}) = \begin{cases}
+ g \;\eta_s^{(b)}(\tau,t_{n}) \;\rho(\tau)_{k=1}) / \rho_o & \text{non-linear case} \\
+ g \;\eta_s^{(b)}(\tau,t_{n}) & \text{linear case}
+ \end{cases}
+\end{equation}
+To get started, we assume the following initial conditions
+\begin{equation} \label{DYN_spg_ts_eta}
+\begin{split}
+\eta^{(b)}(\tau,t_{n=0}) &= \overline{\eta^{(b)}(\tau)}
+\\
+\eta^{(b)}(\tau,t_{n=1}) &= \eta^{(b)}(\tau,t_{n=0}) + \rdt \ \text{RHS}_{n=0}
+\end{split}
+\end{equation}
+with
+\begin{equation} \label{DYN_spg_ts_etaF}
+ \overline{\eta^{(b)}(\tau)} = \frac{1}{N+1} \sum\limits_{n=0}^N \eta^{(b)}(\tau-\rdt,t_{n})
+\end{equation}
+the time averaged surface height taken from the previous barotropic cycle. Likewise,
+\begin{equation} \label{DYN_spg_ts_u}
+\textbf{U}^{(b)}(\tau,t_{n=0}) = \overline{\textbf{U}^{(b)}(\tau)} \\
+\\
+\textbf{U}(\tau,t_{n=1}) = \textbf{U}^{(b)}(\tau,t_{n=0}) + \rdt \ \text{RHS}_{n=0}
+\end{equation}
+with
+\begin{equation} \label{DYN_spg_ts_u}
+ \overline{\textbf{U}^{(b)}(\tau)}
+ = \frac{1}{N+1} \sum\limits_{n=0}^N\textbf{U}^{(b)}(\tau-\rdt,t_{n})
+\end{equation}
+the time averaged vertically integrated transport. Notably, there is no Robert-Asselin time filter used in the barotropic portion of the integration.
+
+Upon reaching $t_{n=N} = \tau + 2\rdt \tau$ , the vertically integrated velocity is time averaged to produce the updated vertically integrated velocity at baroclinic time $\tau + \rdt \tau$
+\begin{equation} \label{DYN_spg_ts_u}
+\textbf{U}(\tau+\rdt) = \overline{\textbf{U}^{(b)}(\tau+\rdt)}
+ = \frac{1}{N+1} \sum\limits_{n=0}^N\textbf{U}^{(b)}(\tau,t_{n})
+\end{equation}
+The surface height on the new baroclinic time step is then determined via a baroclinic leap-frog using the following form
+
+\begin{equation} \label{DYN_spg_ts_ssh}
+\eta(\tau+\Delta) - \eta^{F}(\tau-\Delta) = 2\rdt \ \left[ - \nabla \cdot \textbf{U}(\tau) + \text{EMP}_w \right]
+\end{equation}
+
+ The use of this "big-leap-frog" scheme for the surface height ensures compatibility between the mass/volume budgets and the tracer budgets. More discussion of this point is provided in Chapter 10 (see in particular Section 10.2).
+
+In general, some form of time filter is needed to maintain integrity of the surface
+height field due to the leap-frog splitting mode in equation \ref{DYN_spg_ts_ssh}. We
+have tried various forms of such filtering, with the following method discussed in
+\cite{Griffies_al_MWR01} chosen due to its stability and reasonably good maintenance of
+tracer conservation properties (see Section ??)
+
+\begin{equation} \label{DYN_spg_ts_sshf}
+\eta^{F}(\tau-\Delta) = \overline{\eta^{(b)}(\tau)}
+\end{equation}
+Another approach tried was
+
+\begin{equation} \label{DYN_spg_ts_sshf2}
+\eta^{F}(\tau-\Delta) = \eta(\tau)
+ + (\alpha/2) \left[\overline{\eta^{(b)}}(\tau+\rdt)
+ + \overline{\eta^{(b)}}(\tau-\rdt) -2 \;\eta(\tau) \right]
+\end{equation}
+
+which is useful since it isolates all the time filtering aspects into the term multiplied
+by $\alpha$. This isolation allows for an easy check that tracer conservation is exact when
+eliminating tracer and surface height time filtering (see Section ?? for more complete discussion). However, in the general case with a non-zero $\alpha$, the filter \ref{DYN_spg_ts_sshf} was found to be more conservative, and so is recommended.
+
+} %%end gm comment (copy of griffies book)
+
+%>>>>>===============
+
+
+%--------------------------------------------------------------------------------------------------------------
+% Filtered free surface formulation
+%--------------------------------------------------------------------------------------------------------------
+\subsection{Filtered free surface (\key{dynspg\_flt})}
+\label{DYN_spg_fltp}
+
+The filtered formulation follows the \citet{Roullet_Madec_JGR00} implementation.
+The extra term introduced in the equations (see \S\ref{PE_free_surface}) is solved implicitly.
+The elliptic solvers available in the code are documented in \S\ref{MISC}.
+
+%% gm %%======>>>> given here the discrete eqs provided to the solver
+\gmcomment{ %%% copy from chap-model basics
+\begin{equation} \label{Eq_spg_flt}
+\frac{\partial {\rm {\bf U}}_h }{\partial t}= {\rm {\bf M}}
+- g \nabla \left( \tilde{\rho} \ \eta \right)
+- g \ T_c \nabla \left( \widetilde{\rho} \ \partial_t \eta \right)
+\end{equation}
+where $T_c$, is a parameter with dimensions of time which characterizes the force,
+$\widetilde{\rho} = \rho / \rho_o$ is the dimensionless density, and $\rm {\bf M}$
+represents the collected contributions of the Coriolis, hydrostatic pressure gradient,
+non-linear and viscous terms in \eqref{Eq_PE_dyn}.
+} %end gmcomment
+
+Note that in the linear free surface formulation (\key{vvl} not defined), the ocean depth
+is time-independent and so is the matrix to be inverted. It is computed once and for all and applies to all ocean time steps.
+
+% ================================================================
+% Lateral diffusion term
+% ================================================================
+\section [Lateral diffusion term (\textit{dynldf})]
+ {Lateral diffusion term (\mdl{dynldf})}
+\label{DYN_ldf}
+%------------------------------------------nam_dynldf----------------------------------------------------
+\namdisplay{namdyn_ldf}
+%-------------------------------------------------------------------------------------------------------------
+
+Options are defined through the \ngn{namdyn\_ldf} namelist variables.
+The options available for lateral diffusion are to use either laplacian
+(rotated or not) or biharmonic operators. The coefficients may be constant
+or spatially variable; the description of the coefficients is found in the chapter
+on lateral physics (Chap.\ref{LDF}). The lateral diffusion of momentum is
+evaluated using a forward scheme, $i.e.$ the velocity appearing in its expression
+is the \textit{before} velocity in time, except for the pure vertical component
+that appears when a tensor of rotation is used. This latter term is solved
+implicitly together with the vertical diffusion term (see \S\ref{STP})
+
+At the lateral boundaries either free slip, no slip or partial slip boundary
+conditions are applied according to the user's choice (see Chap.\ref{LBC}).
+
+% ================================================================
+\subsection [Iso-level laplacian operator (\np{ln\_dynldf\_lap}) ]
+ {Iso-level laplacian operator (\np{ln\_dynldf\_lap}=true)}
+\label{DYN_ldf_lap}
+
+For lateral iso-level diffusion, the discrete operator is:
+\begin{equation} \label{Eq_dynldf_lap}
+\left\{ \begin{aligned}
+ D_u^{l{\rm {\bf U}}} =\frac{1}{e_{1u} }\delta _{i+1/2} \left[ {A_T^{lm}
+\;\chi } \right]-\frac{1}{e_{2u} {\kern 1pt}e_{3u} }\delta _j \left[
+{A_f^{lm} \;e_{3f} \zeta } \right] \\
+\\
+ D_v^{l{\rm {\bf U}}} =\frac{1}{e_{2v} }\delta _{j+1/2} \left[ {A_T^{lm}
+\;\chi } \right]+\frac{1}{e_{1v} {\kern 1pt}e_{3v} }\delta _i \left[
+{A_f^{lm} \;e_{3f} \zeta } \right] \\
+\end{aligned} \right.
+\end{equation}
+
+As explained in \S\ref{PE_ldf}, this formulation (as the gradient of a divergence
+and curl of the vorticity) preserves symmetry and ensures a complete
+separation between the vorticity and divergence parts of the momentum diffusion.
+
+%--------------------------------------------------------------------------------------------------------------
+% Rotated laplacian operator
+%--------------------------------------------------------------------------------------------------------------
+\subsection [Rotated laplacian operator (\np{ln\_dynldf\_iso}) ]
+ {Rotated laplacian operator (\np{ln\_dynldf\_iso}=true)}
+\label{DYN_ldf_iso}
+
+A rotation of the lateral momentum diffusion operator is needed in several cases:
+for iso-neutral diffusion in the $z$-coordinate (\np{ln\_dynldf\_iso}=true) and for
+either iso-neutral (\np{ln\_dynldf\_iso}=true) or geopotential
+(\np{ln\_dynldf\_hor}=true) diffusion in the $s$-coordinate. In the partial step
+case, coordinates are horizontal except at the deepest level and no
+rotation is performed when \np{ln\_dynldf\_hor}=true. The diffusion operator
+is defined simply as the divergence of down gradient momentum fluxes on each
+momentum component. It must be emphasized that this formulation ignores
+constraints on the stress tensor such as symmetry. The resulting discrete
+representation is:
+\begin{equation} \label{Eq_dyn_ldf_iso}
+\begin{split}
+ D_u^{l\textbf{U}} &= \frac{1}{e_{1u} \, e_{2u} \, e_{3u} } \\
+& \left\{\quad {\delta _{i+1/2} \left[ {A_T^{lm} \left(
+ {\frac{e_{2t} \; e_{3t} }{e_{1t} } \,\delta _{i}[u]
+ -e_{2t} \; r_{1t} \,\overline{\overline {\delta _{k+1/2}[u]}}^{\,i,\,k}}
+ \right)} \right]} \right.
+\\
+& \qquad +\ \delta_j \left[ {A_f^{lm} \left( {\frac{e_{1f}\,e_{3f} }{e_{2f}
+}\,\delta _{j+1/2} [u] - e_{1f}\, r_{2f}
+\,\overline{\overline {\delta _{k+1/2} [u]}} ^{\,j+1/2,\,k}}
+\right)} \right]
+\\
+&\qquad +\ \delta_k \left[ {A_{uw}^{lm} \left( {-e_{2u} \, r_{1uw} \,\overline{\overline
+{\delta_{i+1/2} [u]}}^{\,i+1/2,\,k+1/2} }
+\right.} \right.
+\\
+& \ \qquad \qquad \qquad \quad\
+- e_{1u} \, r_{2uw} \,\overline{\overline {\delta_{j+1/2} [u]}} ^{\,j,\,k+1/2}
+\\
+& \left. {\left. { \ \qquad \qquad \qquad \ \ \ \left. {\
++\frac{e_{1u}\, e_{2u} }{e_{3uw} }\,\left( {r_{1uw}^2+r_{2uw}^2}
+\right)\,\delta_{k+1/2} [u]} \right)} \right]\;\;\;} \right\}
+\\
+\\
+ D_v^{l\textbf{V}} &= \frac{1}{e_{1v} \, e_{2v} \, e_{3v} } \\
+& \left\{\quad {\delta _{i+1/2} \left[ {A_f^{lm} \left(
+ {\frac{e_{2f} \; e_{3f} }{e_{1f} } \,\delta _{i+1/2}[v]
+ -e_{2f} \; r_{1f} \,\overline{\overline {\delta _{k+1/2}[v]}}^{\,i+1/2,\,k}}
+ \right)} \right]} \right.
+\\
+& \qquad +\ \delta_j \left[ {A_T^{lm} \left( {\frac{e_{1t}\,e_{3t} }{e_{2t}
+}\,\delta _{j} [v] - e_{1t}\, r_{2t}
+\,\overline{\overline {\delta _{k+1/2} [v]}} ^{\,j,\,k}}
+\right)} \right]
+\\
+& \qquad +\ \delta_k \left[ {A_{vw}^{lm} \left( {-e_{2v} \, r_{1vw} \,\overline{\overline
+{\delta_{i+1/2} [v]}}^{\,i+1/2,\,k+1/2} }\right.} \right.
+\\
+& \ \qquad \qquad \qquad \quad\
+- e_{1v} \, r_{2vw} \,\overline{\overline {\delta_{j+1/2} [v]}} ^{\,j+1/2,\,k+1/2}
+\\
+& \left. {\left. { \ \qquad \qquad \qquad \ \ \ \left. {\
++\frac{e_{1v}\, e_{2v} }{e_{3vw} }\,\left( {r_{1vw}^2+r_{2vw}^2}
+\right)\,\delta_{k+1/2} [v]} \right)} \right]\;\;\;} \right\}
+ \end{split}
+\end{equation}
+where $r_1$ and $r_2$ are the slopes between the surface along which the
+diffusion operator acts and the surface of computation ($z$- or $s$-surfaces).
+The way these slopes are evaluated is given in the lateral physics chapter
+(Chap.\ref{LDF}).
+
+%--------------------------------------------------------------------------------------------------------------
+% Iso-level bilaplacian operator
+%--------------------------------------------------------------------------------------------------------------
+\subsection [Iso-level bilaplacian operator (\np{ln\_dynldf\_bilap})]
+ {Iso-level bilaplacian operator (\np{ln\_dynldf\_bilap}=true)}
+\label{DYN_ldf_bilap}
+
+The lateral fourth order operator formulation on momentum is obtained by
+applying \eqref{Eq_dynldf_lap} twice. It requires an additional assumption on
+boundary conditions: the first derivative term normal to the coast depends on
+the free or no-slip lateral boundary conditions chosen, while the third
+derivative terms normal to the coast are set to zero (see Chap.\ref{LBC}).
+%%%
+\gmcomment{add a remark on the the change in the position of the coefficient}
+%%%
+
+% ================================================================
+% Vertical diffusion term
+% ================================================================
+\section [Vertical diffusion term (\mdl{dynzdf})]
+ {Vertical diffusion term (\mdl{dynzdf})}
+\label{DYN_zdf}
+%----------------------------------------------namzdf------------------------------------------------------
+\namdisplay{namzdf}
+%-------------------------------------------------------------------------------------------------------------
+
+Options are defined through the \ngn{namzdf} namelist variables.
+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. Two time stepping schemes
+can be used for the vertical diffusion term : $(a)$ a forward time differencing
+scheme (\np{ln\_zdfexp}=true) using a time splitting technique
+(\np{nn\_zdfexp} $>$ 1) or $(b)$ a backward (or implicit) time differencing scheme
+(\np{ln\_zdfexp}=false) (see \S\ref{STP}). Note that namelist variables
+\np{ln\_zdfexp} and \np{nn\_zdfexp} apply to both tracers and dynamics.
+
+The formulation of the vertical subgrid scale physics is the same whatever
+the vertical coordinate is. The vertical diffusion operators given by
+\eqref{Eq_PE_zdf} take the following semi-discrete space form:
+\begin{equation} \label{Eq_dynzdf}
+\left\{ \begin{aligned}
+D_u^{vm} &\equiv \frac{1}{e_{3u}} \ \delta _k \left[ \frac{A_{uw}^{vm} }{e_{3uw} }
+ \ \delta _{k+1/2} [\,u\,] \right] \\
+\\
+D_v^{vm} &\equiv \frac{1}{e_{3v}} \ \delta _k \left[ \frac{A_{vw}^{vm} }{e_{3vw} }
+ \ \delta _{k+1/2} [\,v\,] \right]
+\end{aligned} \right.
+\end{equation}
+where $A_{uw}^{vm} $ and $A_{vw}^{vm} $ are the vertical eddy viscosity and
+diffusivity coefficients. The way these coefficients are evaluated
+depends on the vertical physics used (see \S\ref{ZDF}).
+
+The surface boundary condition on momentum is the stress exerted by
+the wind. At the surface, the momentum fluxes are prescribed as the boundary
+condition on the vertical turbulent momentum fluxes,
+\begin{equation} \label{Eq_dynzdf_sbc}
+\left.{\left( {\frac{A^{vm} }{e_3 }\ \frac{\partial \textbf{U}_h}{\partial k}} \right)} \right|_{z=1}
+ = \frac{1}{\rho _o} \binom{\tau _u}{\tau _v }
+\end{equation}
+where $\left( \tau _u ,\tau _v \right)$ are the two components of the wind stress
+vector in the (\textbf{i},\textbf{j}) coordinate system. The high mixing coefficients
+in the surface mixed layer ensure that the surface wind stress is distributed in
+the vertical over the mixed layer depth. If the vertical mixing coefficient
+is small (when no mixed layer scheme is used) the surface stress enters only
+the top model level, as a body force. The surface wind stress is calculated
+in the surface module routines (SBC, see Chap.\ref{SBC})
+
+The turbulent flux of momentum at the bottom of the ocean is specified through
+a bottom friction parameterisation (see \S\ref{ZDF_bfr})
+
+% ================================================================
+% External Forcing
+% ================================================================
+\section{External Forcings}
+\label{DYN_forcing}
+
+Besides the surface and bottom stresses (see the above section) which are
+introduced as boundary conditions on the vertical mixing, two other forcings
+enter the dynamical equations.
+
+One is the effect of atmospheric pressure on the ocean dynamics.
+Another forcing term is the tidal potential.
+Both of which will be introduced into the reference version soon.
+
+\gmcomment{atmospheric pressure is there!!!! include its description }
+
+% ================================================================
+% Time evolution term
+% ================================================================
+\section [Time evolution term (\textit{dynnxt})]
+ {Time evolution term (\mdl{dynnxt})}
+\label{DYN_nxt}
+
+%----------------------------------------------namdom----------------------------------------------------
+\namdisplay{namdom}
+%-------------------------------------------------------------------------------------------------------------
+
+Options are defined through the \ngn{namdom} namelist variables.
+The general framework for dynamics time stepping is a leap-frog scheme,
+$i.e.$ a three level centred time scheme associated with an Asselin time filter
+(cf. Chap.\ref{STP}). The scheme is applied to the velocity, except when using
+the flux form of momentum advection (cf. \S\ref{DYN_adv_cor_flux}) in the variable
+volume case (\key{vvl} defined), where it has to be applied to the thickness
+weighted velocity (see \S\ref{Apdx_A_momentum})
+
+$\bullet$ vector invariant form or linear free surface (\np{ln\_dynhpg\_vec}=true ; \key{vvl} not defined):
+\begin{equation} \label{Eq_dynnxt_vec}
+\left\{ \begin{aligned}
+&u^{t+\rdt} = u_f^{t-\rdt} + 2\rdt \ \text{RHS}_u^t \\
+&u_f^t \;\quad = u^t+\gamma \,\left[ {u_f^{t-\rdt} -2u^t+u^{t+\rdt}} \right]
+\end{aligned} \right.
+\end{equation}
+
+$\bullet$ flux form and nonlinear free surface (\np{ln\_dynhpg\_vec}=false ; \key{vvl} defined):
+\begin{equation} \label{Eq_dynnxt_flux}
+\left\{ \begin{aligned}
+&\left(e_{3u}\,u\right)^{t+\rdt} = \left(e_{3u}\,u\right)_f^{t-\rdt} + 2\rdt \; e_{3u} \;\text{RHS}_u^t \\
+&\left(e_{3u}\,u\right)_f^t \;\quad = \left(e_{3u}\,u\right)^t
+ +\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]
+\end{aligned} \right.
+\end{equation}
+where RHS is the right hand side of the momentum equation, the subscript $f$
+denotes filtered values and $\gamma$ is the Asselin coefficient. $\gamma$ is
+initialized as \np{nn\_atfp} (namelist parameter). Its default value is \np{nn\_atfp} = $10^{-3}$.
+In both cases, the modified Asselin filter is not applied since perfect conservation
+is not an issue for the momentum equations.
+
+Note that with the filtered free surface, the update of the \textit{after} velocities
+is done in the \mdl{dynsp\_flt} module, and only array swapping
+and Asselin filtering is done in \mdl{dynnxt}.
+
+% ================================================================
+% Neptune effect
+% ================================================================
+\section [Neptune effect (\textit{dynnept})]
+ {Neptune effect (\mdl{dynnept})}
+\label{DYN_nept}
+
+The "Neptune effect" (thus named in \citep{HollowayOM86}) is a
+parameterisation of the potentially large effect of topographic form stress
+(caused by eddies) in driving the ocean circulation. Originally developed for
+low-resolution models, in which it was applied via a Laplacian (second-order)
+diffusion-like term in the momentum equation, it can also be applied in eddy
+permitting or resolving models, in which a more scale-selective bilaplacian
+(fourth-order) implementation is preferred. This mechanism has a
+significant effect on boundary currents (including undercurrents), and the
+upwelling of deep water near continental shelves.
+
+The theoretical basis for the method can be found in
+\citep{HollowayJPO92}, including the explanation of why form stress is not
+necessarily a drag force, but may actually drive the flow.
+\citep{HollowayJPO94} demonstrate the effects of the parameterisation in
+the GFDL-MOM model, at a horizontal resolution of about 1.8 degrees.
+\citep{HollowayOM08} demonstrate the biharmonic version of the
+parameterisation in a global run of the POP model, with an average horizontal
+grid spacing of about 32km.
+
+The NEMO implementation is a simplified form of that supplied by
+Greg Holloway, the testing of which was described in \citep{HollowayJGR09}.
+The major simplification is that a time invariant Neptune velocity
+field is assumed. This is computed only once, during start-up, and
+made available to the rest of the code via a module. Vertical
+diffusive terms are also ignored, and the model topography itself
+is used, rather than a separate topographic dataset as in
+\citep{HollowayOM08}. This implementation is only in the iso-level
+formulation, as is the case anyway for the bilaplacian operator.
+
+The velocity field is derived from a transport stream function given by:
+
+\begin{equation} \label{Eq_dynnept_sf}
+\psi = -fL^2H
+\end{equation}
+
+where $L$ is a latitude-dependant length scale given by:
+
+\begin{equation} \label{Eq_dynnept_ls}
+L = l_1 + (l_2 -l_1)\left ( {1 + \cos 2\phi \over 2 } \right )
+\end{equation}
+
+where $\phi$ is latitude and $l_1$ and $l_2$ are polar and equatorial length scales respectively.
+Neptune velocity components, $u^*$, $v^*$ are derived from the stremfunction as:
+
+\begin{equation} \label{Eq_dynnept_vel}
+u^* = -{1\over H} {\partial \psi \over \partial y}\ \ \ ,\ \ \ v^* = {1\over H} {\partial \psi \over \partial x}
+\end{equation}
+
+\smallskip
+%----------------------------------------------namdom----------------------------------------------------
+\namdisplay{namdyn_nept}
+%--------------------------------------------------------------------------------------------------------
+\smallskip
+
+The Neptune effect is enabled when \np{ln\_neptsimp}=true (default=false).
+\np{ln\_smooth\_neptvel} controls whether a scale-selective smoothing is applied
+to the Neptune effect flow field (default=false) (this smoothing method is as
+used by Holloway). \np{rn\_tslse} and \np{rn\_tslsp} are the equatorial and
+polar values respectively of the length-scale parameter $L$ used in determining
+the Neptune stream function \eqref{Eq_dynnept_sf} and \eqref{Eq_dynnept_ls}.
+Values at intermediate latitudes are given by a cosine fit, mimicking the
+variation of the deformation radius with latitude. The default values of 12km
+and 3km are those given in \citep{HollowayJPO94}, appropriate for a coarse
+resolution model. The finer resolution study of \citep{HollowayOM08} increased
+the values of L by a factor of $\sqrt 2$ to 17km and 4.2km, thus doubling the
+stream function for a given topography.
+
+The simple formulation for ($u^*$, $v^*$) can give unacceptably large velocities
+in shallow water, and \citep{HollowayOM08} add an offset to the depth in the
+denominator to control this problem. In this implementation we offer instead (at
+the suggestion of G. Madec) the option of ramping down the Neptune flow field to
+zero over a finite depth range. The switch \np{ln\_neptramp} activates this
+option (default=false), in which case velocities at depths greater than
+\np{rn\_htrmax} are unaltered, but ramp down linearly with depth to zero at a
+depth of \np{rn\_htrmin} (and shallower).
+
+% ================================================================
Index: /branches/UKMO/dev_r5518_GO6_under_ice_relax_dr_hook/DOC/TexFiles/Chapters/Chap_LBC.tex
===================================================================
--- /branches/UKMO/dev_r5518_GO6_under_ice_relax_dr_hook/DOC/TexFiles/Chapters/Chap_LBC.tex (revision 11643)
+++ /branches/UKMO/dev_r5518_GO6_under_ice_relax_dr_hook/DOC/TexFiles/Chapters/Chap_LBC.tex (revision 11643)
@@ -0,0 +1,1035 @@
+% ================================================================
+% Chapter � Lateral Boundary Condition (LBC)
+% ================================================================
+\chapter{Lateral Boundary Condition (LBC) }
+\label{LBC}
+\minitoc
+
+\newpage
+$\ $\newline % force a new ligne
+
+
+%gm% add here introduction to this chapter
+
+% ================================================================
+% Boundary Condition at the Coast
+% ================================================================
+\section{Boundary Condition at the Coast (\np{rn\_shlat})}
+\label{LBC_coast}
+%--------------------------------------------nam_lbc-------------------------------------------------------
+\namdisplay{namlbc}
+%--------------------------------------------------------------------------------------------------------------
+
+%The lateral ocean boundary conditions contiguous to coastlines are Neumann conditions for heat and salt (no flux across boundaries) and Dirichlet conditions for momentum (ranging from free-slip to "strong" no-slip). They are handled automatically by the mask system (see \S\ref{DOM_msk}).
+
+%OPA allows land and topography grid points in the computational domain due to the presence of continents or islands, and includes the use of a full or partial step representation of bottom topography. The computation is performed over the whole domain, i.e. we do not try to restrict the computation to ocean-only points. This choice has two motivations. Firstly, working on ocean only grid points overloads the code and harms the code readability. Secondly, and more importantly, it drastically reduces the vector portion of the computation, leading to a dramatic increase of CPU time requirement on vector computers. The current section describes how the masking affects the computation of the various terms of the equations with respect to the boundary condition at solid walls. The process of defining which areas are to be masked is described in \S\ref{DOM_msk}.
+
+Options are defined through the \ngn{namlbc} namelist variables.
+The discrete representation of a domain with complex boundaries (coastlines and
+bottom topography) leads to arrays that include large portions where a computation
+is not required as the model variables remain at zero. Nevertheless, vectorial
+supercomputers are far more efficient when computing over a whole array, and the
+readability of a code is greatly improved when boundary conditions are applied in
+an automatic way rather than by a specific computation before or after each
+computational loop. An efficient way to work over the whole domain while specifying
+the boundary conditions, is to use multiplication by mask arrays in the computation.
+A mask array is a matrix whose elements are $1$ in the ocean domain and $0$
+elsewhere. A simple multiplication of a variable by its own mask ensures that it will
+remain zero over land areas. Since most of the boundary conditions consist of a
+zero flux across the solid boundaries, they can be simply applied by multiplying
+variables by the correct mask arrays, $i.e.$ the mask array of the grid point where
+the flux is evaluated. For example, the heat flux in the \textbf{i}-direction is evaluated
+at $u$-points. Evaluating this quantity as,
+
+\begin{equation} \label{Eq_lbc_aaaa}
+\frac{A^{lT} }{e_1 }\frac{\partial T}{\partial i}\equiv \frac{A_u^{lT}
+}{e_{1u} } \; \delta _{i+1 / 2} \left[ T \right]\;\;mask_u
+\end{equation}
+(where mask$_{u}$ is the mask array at a $u$-point) ensures that the heat flux is
+zero inside land and at the boundaries, since mask$_{u}$ is zero at solid boundaries
+which in this case are defined at $u$-points (normal velocity $u$ remains zero at
+the coast) (Fig.~\ref{Fig_LBC_uv}).
+
+%>>>>>>>>>>>>>>>>>>>>>>>>>>>>
+\begin{figure}[!t] \begin{center}
+\includegraphics[width=0.90\textwidth]{./TexFiles/Figures/Fig_LBC_uv.pdf}
+\caption{ \label{Fig_LBC_uv}
+Lateral boundary (thick line) at T-level. The velocity normal to the boundary is set to zero.}
+\end{center} \end{figure}
+%>>>>>>>>>>>>>>>>>>>>>>>>>>>>
+
+For momentum the situation is a bit more complex as two boundary conditions
+must be provided along the coast (one each for the normal and tangential velocities).
+The boundary of the ocean in the C-grid is defined by the velocity-faces.
+For example, at a given $T$-level, the lateral boundary (a coastline or an intersection
+with the bottom topography) is made of segments joining $f$-points, and normal
+velocity points are located between two $f-$points (Fig.~\ref{Fig_LBC_uv}).
+The boundary condition on the normal velocity (no flux through solid boundaries)
+can thus be easily implemented using the mask system. The boundary condition
+on the tangential velocity requires a more specific treatment. This boundary
+condition influences the relative vorticity and momentum diffusive trends, and is
+required in order to compute the vorticity at the coast. Four different types of
+lateral boundary condition are available, controlled by the value of the \np{rn\_shlat}
+namelist parameter. (The value of the mask$_{f}$ array along the coastline is set
+equal to this parameter.) These are:
+
+%>>>>>>>>>>>>>>>>>>>>>>>>>>>>
+\begin{figure}[!p] \begin{center}
+\includegraphics[width=0.90\textwidth]{./TexFiles/Figures/Fig_LBC_shlat.pdf}
+\caption{ \label{Fig_LBC_shlat}
+lateral boundary condition (a) free-slip ($rn\_shlat=0$) ; (b) no-slip ($rn\_shlat=2$)
+; (c) "partial" free-slip ($0>>>>>>>>>>>>>>>>>>>>>>>>>>>
+
+\begin{description}
+
+\item[free-slip boundary condition (\np{rn\_shlat}=0): ] the tangential velocity at the
+coastline is equal to the offshore velocity, $i.e.$ the normal derivative of the
+tangential velocity is zero at the coast, so the vorticity: mask$_{f}$ array is set
+to zero inside the land and just at the coast (Fig.~\ref{Fig_LBC_shlat}-a).
+
+\item[no-slip boundary condition (\np{rn\_shlat}=2): ] the tangential velocity vanishes
+at the coastline. Assuming that the tangential velocity decreases linearly from
+the closest ocean velocity grid point to the coastline, the normal derivative is
+evaluated as if the velocities at the closest land velocity gridpoint and the closest
+ocean velocity gridpoint were of the same magnitude but in the opposite direction
+(Fig.~\ref{Fig_LBC_shlat}-b). Therefore, the vorticity along the coastlines is given by:
+
+\begin{equation*}
+\zeta \equiv 2 \left(\delta_{i+1/2} \left[e_{2v} v \right] - \delta_{j+1/2} \left[e_{1u} u \right] \right) / \left(e_{1f} e_{2f} \right) \ ,
+\end{equation*}
+where $u$ and $v$ are masked fields. Setting the mask$_{f}$ array to $2$ along
+the coastline provides a vorticity field computed with the no-slip boundary condition,
+simply by multiplying it by the mask$_{f}$ :
+\begin{equation} \label{Eq_lbc_bbbb}
+\zeta \equiv \frac{1}{e_{1f} {\kern 1pt}e_{2f} }\left( {\delta _{i+1/2}
+\left[ {e_{2v} \,v} \right]-\delta _{j+1/2} \left[ {e_{1u} \,u} \right]}
+\right)\;\mbox{mask}_f
+\end{equation}
+
+\item["partial" free-slip boundary condition (0$<$\np{rn\_shlat}$<$2): ] the tangential
+velocity at the coastline is smaller than the offshore velocity, $i.e.$ there is a lateral
+friction but not strong enough to make the tangential velocity at the coast vanish
+(Fig.~\ref{Fig_LBC_shlat}-c). This can be selected by providing a value of mask$_{f}$
+strictly inbetween $0$ and $2$.
+
+\item["strong" no-slip boundary condition (2$<$\np{rn\_shlat}): ] the viscous boundary
+layer is assumed to be smaller than half the grid size (Fig.~\ref{Fig_LBC_shlat}-d).
+The friction is thus larger than in the no-slip case.
+
+\end{description}
+
+Note that when the bottom topography is entirely represented by the $s$-coor-dinates
+(pure $s$-coordinate), the lateral boundary condition on tangential velocity is of much
+less importance as it is only applied next to the coast where the minimum water depth
+can be quite shallow.
+
+The alternative numerical implementation of the no-slip boundary conditions for an
+arbitrary coast line of \citet{Shchepetkin1996} is also available through the
+\key{noslip\_accurate} CPP key. It is based on a fourth order evaluation of the shear at the
+coast which, in turn, allows a true second order scheme in the interior of the domain
+($i.e.$ the numerical boundary scheme simulates the truncation error of the numerical
+scheme used in the interior of the domain). \citet{Shchepetkin1996} found that such a
+technique considerably improves the quality of the numerical solution. In \NEMO, such
+spectacular improvements have not been found in the half-degree global ocean
+(ORCA05), but significant reductions of numerically induced coastal upwellings were
+found in an eddy resolving simulation of the Alboran Sea \citep{Olivier_PhD01}.
+Nevertheless, since a no-slip boundary condition is not recommended in an eddy
+permitting or resolving simulation \citep{Penduff_al_OS07}, the use of this option is also
+not recommended.
+
+In practice, the no-slip accurate option changes the way the curl is evaluated at the
+coast (see \mdl{divcur} module), and requires the nature of each coastline grid point
+(convex or concave corners, straight north-south or east-west coast) to be specified.
+This is performed in routine \rou{dom\_msk\_nsa} in the \mdl{domask} module.
+
+% ================================================================
+% Boundary Condition around the Model Domain
+% ================================================================
+\section{Model Domain Boundary Condition (\np{jperio})}
+\label{LBC_jperio}
+
+At the model domain boundaries several choices are offered: closed, cyclic east-west,
+south symmetric across the equator, a north-fold, and combination closed-north fold
+or cyclic-north-fold. The north-fold boundary condition is associated with the 3-pole ORCA mesh.
+
+% -------------------------------------------------------------------------------------------------------------
+% Closed, cyclic, south symmetric (\np{jperio} = 0, 1 or 2)
+% -------------------------------------------------------------------------------------------------------------
+\subsection{Closed, cyclic, south symmetric (\np{jperio} = 0, 1 or 2)}
+\label{LBC_jperio012}
+
+The choice of closed, cyclic or symmetric model domain boundary condition is made
+by setting \np{jperio} to 0, 1 or 2 in namelist \ngn{namcfg}. Each time such a boundary
+condition is needed, it is set by a call to routine \mdl{lbclnk}. The computation of
+momentum and tracer trends proceeds from $i=2$ to $i=jpi-1$ and from $j=2$ to
+$j=jpj-1$, $i.e.$ in the model interior. To choose a lateral model boundary condition
+is to specify the first and last rows and columns of the model variables.
+
+\begin{description}
+
+\item[For closed boundary (\textit{jperio=0})], solid walls are imposed at all model
+boundaries: first and last rows and columns are set to zero.
+
+\item[For cyclic east-west boundary (\textit{jperio=1})], first and last rows are set
+to zero (closed) whilst the first column is set to the value of the last-but-one column
+and the last column to the value of the second one (Fig.~\ref{Fig_LBC_jperio}-a).
+Whatever flows out of the eastern (western) end of the basin enters the western
+(eastern) end. Note that there is no option for north-south cyclic or for doubly
+cyclic cases.
+
+\item[For symmetric boundary condition across the equator (\textit{jperio=2})],
+last rows, and first and last columns are set to zero (closed). The row of symmetry
+is chosen to be the $u$- and $T-$points equator line ($j=2$, i.e. at the southern
+end of the domain). For arrays defined at $u-$ or $T-$points, the first row is set
+to the value of the third row while for most of $v$- and $f$-point arrays ($v$, $\zeta$,
+$j\psi$, but \gmcomment{not sure why this is "but"} scalar arrays such as eddy coefficients)
+the first row is set to minus the value of the second row (Fig.~\ref{Fig_LBC_jperio}-b).
+Note that this boundary condition is not yet available for the case of a massively
+parallel computer (\textbf{key{\_}mpp} defined).
+
+\end{description}
+
+%>>>>>>>>>>>>>>>>>>>>>>>>>>>>
+\begin{figure}[!t] \begin{center}
+\includegraphics[width=1.0\textwidth]{./TexFiles/Figures/Fig_LBC_jperio.pdf}
+\caption{ \label{Fig_LBC_jperio}
+setting of (a) east-west cyclic (b) symmetric across the equator boundary conditions.}
+\end{center} \end{figure}
+%>>>>>>>>>>>>>>>>>>>>>>>>>>>>
+
+% -------------------------------------------------------------------------------------------------------------
+% North fold (\textit{jperio = 3 }to $6)$
+% -------------------------------------------------------------------------------------------------------------
+\subsection{North-fold (\textit{jperio = 3 }to $6)$ }
+\label{LBC_north_fold}
+
+The north fold boundary condition has been introduced in order to handle the north
+boundary of a three-polar ORCA grid. Such a grid has two poles in the northern hemisphere.
+\colorbox{yellow}{to be completed...}
+
+%>>>>>>>>>>>>>>>>>>>>>>>>>>>>
+\begin{figure}[!t] \begin{center}
+\includegraphics[width=0.90\textwidth]{./TexFiles/Figures/Fig_North_Fold_T.pdf}
+\caption{ \label{Fig_North_Fold_T}
+North fold boundary with a $T$-point pivot and cyclic east-west boundary condition
+($jperio=4$), as used in ORCA 2, 1/4, and 1/12. Pink shaded area corresponds
+to the inner domain mask (see text). }
+\end{center} \end{figure}
+%>>>>>>>>>>>>>>>>>>>>>>>>>>>>
+
+% ====================================================================
+% Exchange with neighbouring processors
+% ====================================================================
+\section [Exchange with neighbouring processors (\textit{lbclnk}, \textit{lib\_mpp})]
+ {Exchange with neighbouring processors (\mdl{lbclnk}, \mdl{lib\_mpp})}
+\label{LBC_mpp}
+
+For massively parallel processing (mpp), a domain decomposition method is used.
+The basic idea of the method is to split the large computation domain of a numerical
+experiment into several smaller domains and solve the set of equations by addressing
+independent local problems. Each processor has its own local memory and computes
+the model equation over a subdomain of the whole model domain. The subdomain
+boundary conditions are specified through communications between processors
+which are organized by explicit statements (message passing method).
+
+A big advantage is that the method does not need many modifications of the initial
+FORTRAN code. From the modeller's point of view, each sub domain running on
+a processor is identical to the "mono-domain" code. In addition, the programmer
+manages the communications between subdomains, and the code is faster when
+the number of processors is increased. The porting of OPA code on an iPSC860
+was achieved during Guyon's PhD [Guyon et al. 1994, 1995] in collaboration with
+CETIIS and ONERA. The implementation in the operational context and the studies
+of performance on a T3D and T3E Cray computers have been made in collaboration
+with IDRIS and CNRS. The present implementation is largely inspired by Guyon's
+work [Guyon 1995].
+
+The parallelization strategy is defined by the physical characteristics of the
+ocean model. Second order finite difference schemes lead to local discrete
+operators that depend at the very most on one neighbouring point. The only
+non-local computations concern the vertical physics (implicit diffusion, 1.5
+turbulent closure scheme, ...) (delocalization over the whole water column),
+and the solving of the elliptic equation associated with the surface pressure
+gradient computation (delocalization over the whole horizontal domain).
+Therefore, a pencil strategy is used for the data sub-structuration
+\gmcomment{no idea what this means!}
+: the 3D initial domain is laid out on local processor
+memories following a 2D horizontal topological splitting. Each sub-domain
+computes its own surface and bottom boundary conditions and has a side
+wall overlapping interface which defines the lateral boundary conditions for
+computations in the inner sub-domain. The overlapping area consists of the
+two rows at each edge of the sub-domain. After a computation, a communication
+phase starts: each processor sends to its neighbouring processors the update
+values of the points corresponding to the interior overlapping area to its
+neighbouring sub-domain (i.e. the innermost of the two overlapping rows).
+The communication is done through message passing. Usually the parallel virtual
+language, PVM, is used as it is a standard language available on nearly all
+MPP computers. More specific languages (i.e. computer dependant languages)
+can be easily used to speed up the communication, such as SHEM on a T3E
+computer. The data exchanges between processors are required at the very
+place where lateral domain boundary conditions are set in the mono-domain
+computation (\S III.10-c): the lbc\_lnk routine which manages such conditions
+is substituted by mpplnk.F or mpplnk2.F routine when running on an MPP
+computer (\key{mpp\_mpi} defined). It has to be pointed out that when using
+the MPP version of the model, the east-west cyclic boundary condition is done
+implicitly, whilst the south-symmetric boundary condition option is not available.
+
+%>>>>>>>>>>>>>>>>>>>>>>>>>>>>
+\begin{figure}[!t] \begin{center}
+\includegraphics[width=0.90\textwidth]{./TexFiles/Figures/Fig_mpp.pdf}
+\caption{ \label{Fig_mpp}
+Positioning of a sub-domain when massively parallel processing is used. }
+\end{center} \end{figure}
+%>>>>>>>>>>>>>>>>>>>>>>>>>>>>
+
+In the standard version of the OPA model, the splitting is regular and arithmetic.
+ the i-axis is divided by \jp{jpni} and the j-axis by \jp{jpnj} for a number of processors
+ \jp{jpnij} most often equal to $jpni \times jpnj$ (model parameters set in
+ \mdl{par\_oce}). Each processor is independent and without message passing
+ or synchronous process
+ \gmcomment{how does a synchronous process relate to this?},
+ programs run alone and access just its own local memory. For this reason, the
+ main model dimensions are now the local dimensions of the subdomain (pencil)
+ that are named \jp{jpi}, \jp{jpj}, \jp{jpk}. These dimensions include the internal
+ domain and the overlapping rows. The number of rows to exchange (known as
+ the halo) is usually set to one (\jp{jpreci}=1, in \mdl{par\_oce}). The whole domain
+ dimensions are named \np{jpiglo}, \np{jpjglo} and \jp{jpk}. The relationship between
+ the whole domain and a sub-domain is:
+\begin{eqnarray}
+ jpi & = & ( jpiglo-2*jpreci + (jpni-1) ) / jpni + 2*jpreci \nonumber \\
+ jpj & = & ( jpjglo-2*jprecj + (jpnj-1) ) / jpnj + 2*jprecj \label{Eq_lbc_jpi}
+\end{eqnarray}
+where \jp{jpni}, \jp{jpnj} are the number of processors following the i- and j-axis.
+
+\colorbox{yellow}{Figure IV.3: example of a domain splitting with 9 processors and
+no east-west cyclic boundary conditions.}
+
+One also defines variables nldi and nlei which correspond to the internal
+domain bounds, and the variables nimpp and njmpp which are the position
+of the (1,1) grid-point in the global domain. An element of $T_{l}$, a local array
+(subdomain) corresponds to an element of $T_{g}$, a global array
+(whole domain) by the relationship:
+\begin{equation} \label{Eq_lbc_nimpp}
+T_{g} (i+nimpp-1,j+njmpp-1,k) = T_{l} (i,j,k),
+\end{equation}
+with $1 \leq i \leq jpi$, $1 \leq j \leq jpj $ , and $1 \leq k \leq jpk$.
+
+Processors are numbered from 0 to $jpnij-1$, the number is saved in the variable
+nproc. In the standard version, a processor has no more than four neighbouring
+processors named nono (for north), noea (east), noso (south) and nowe (west)
+and two variables, nbondi and nbondj, indicate the relative position of the processor
+\colorbox{yellow}{(see Fig.IV.3)}:
+\begin{itemize}
+\item nbondi = -1 an east neighbour, no west processor,
+\item nbondi = 0 an east neighbour, a west neighbour,
+\item nbondi = 1 no east processor, a west neighbour,
+\item nbondi = 2 no splitting following the i-axis.
+\end{itemize}
+During the simulation, processors exchange data with their neighbours.
+If there is effectively a neighbour, the processor receives variables from this
+processor on its overlapping row, and sends the data issued from internal
+domain corresponding to the overlapping row of the other processor.
+
+\colorbox{yellow}{Figure IV.4: pencil splitting with the additional outer halos }
+
+
+The \NEMO model computes equation terms with the help of mask arrays (0 on land
+points and 1 on sea points). It is easily readable and very efficient in the context of
+a computer with vectorial architecture. However, in the case of a scalar processor,
+computations over the land regions become more expensive in terms of CPU time.
+It is worse when we use a complex configuration with a realistic bathymetry like the
+global ocean where more than 50 \% of points are land points. For this reason, a
+pre-processing tool can be used to choose the mpp domain decomposition with a
+maximum number of only land points processors, which can then be eliminated.
+(For example, the mpp\_optimiz tools, available from the DRAKKAR web site.)
+This optimisation is dependent on the specific bathymetry employed. The user
+then chooses optimal parameters \jp{jpni}, \jp{jpnj} and \jp{jpnij} with
+$jpnij < jpni \times jpnj$, leading to the elimination of $jpni \times jpnj - jpnij$
+land processors. When those parameters are specified in module \mdl{par\_oce},
+the algorithm in the \rou{inimpp2} routine sets each processor's parameters (nbound,
+nono, noea,...) so that the land-only processors are not taken into account.
+
+\colorbox{yellow}{Note that the inimpp2 routine is general so that the original inimpp
+routine should be suppressed from the code.}
+
+When land processors are eliminated, the value corresponding to these locations in
+the model output files is zero. Note that this is a problem for a mesh output file written
+by such a model configuration, because model users often divide by the scale factors
+($e1t$, $e2t$, etc) and do not expect the grid size to be zero, even on land. It may be
+best not to eliminate land processors when running the model especially to write the
+mesh files as outputs (when \np{nn\_msh} namelist parameter differs from 0).
+%%
+\gmcomment{Steven : dont understand this, no land processor means no output file
+covering this part of globe; its only when files are stitched together into one that you
+can leave a hole}
+%%
+
+%>>>>>>>>>>>>>>>>>>>>>>>>>>>>
+\begin{figure}[!ht] \begin{center}
+\includegraphics[width=0.90\textwidth]{./TexFiles/Figures/Fig_mppini2.pdf}
+\caption { \label{Fig_mppini2}
+Example of Atlantic domain defined for the CLIPPER projet. Initial grid is
+composed of 773 x 1236 horizontal points.
+(a) the domain is split onto 9 \time 20 subdomains (jpni=9, jpnj=20).
+52 subdomains are land areas.
+(b) 52 subdomains are eliminated (white rectangles) and the resulting number
+of processors really used during the computation is jpnij=128.}
+\end{center} \end{figure}
+%>>>>>>>>>>>>>>>>>>>>>>>>>>>>
+
+
+% ================================================================
+% Open Boundary Conditions
+% ================================================================
+\section{Open Boundary Conditions (\key{obc}) (OBC)}
+\label{LBC_obc}
+%-----------------------------------------nam_obc -------------------------------------------
+%- nobc_dta = 0 ! = 0 the obc data are equal to the initial state
+%- ! = 1 the obc data are read in 'obc .dta' files
+%- rn_dpein = 1. ! ???
+%- rn_dpwin = 1. ! ???
+%- rn_dpnin = 30. ! ???
+%- rn_dpsin = 1. ! ???
+%- rn_dpeob = 1500. ! time relaxation (days) for the east open boundary
+%- rn_dpwob = 15. ! " " for the west open boundary
+%- rn_dpnob = 150. ! " " for the north open boundary
+%- rn_dpsob = 15. ! " " for the south open boundary
+%- ln_obc_clim = .true. ! climatological obc data files (default T)
+%- ln_vol_cst = .true. ! total volume conserved
+\namdisplay{namobc}
+
+It is often necessary to implement a model configuration limited to an oceanic
+region or a basin, which communicates with the rest of the global ocean through
+''open boundaries''. As stated by \citet{Roed1986}, an open boundary is a
+computational border where the aim of the calculations is to allow the perturbations
+generated inside the computational domain to leave it without deterioration of the
+inner model solution. However, an open boundary also has to let information from
+the outer ocean enter the model and should support inflow and outflow conditions.
+
+The open boundary package OBC is the first open boundary option developed in
+NEMO (originally in OPA8.2). It allows the user to
+\begin{itemize}
+\item tell the model that a boundary is ''open'' and not closed by a wall, for example
+by modifying the calculation of the divergence of velocity there;
+\item impose values of tracers and velocities at that boundary (values which may
+be taken from a climatology): this is the``fixed OBC'' option.
+\item calculate boundary values by a sophisticated algorithm combining radiation
+and relaxation (``radiative OBC'' option)
+\end{itemize}
+
+Options are defined through the \ngn{namobc} namelist variables.
+The package resides in the OBC directory. It is described here in four parts: the
+boundary geometry (parameters to be set in \mdl{obc\_par}), the forcing data at
+the boundaries (module \mdl{obcdta}), the radiation algorithm involving the
+namelist and module \mdl{obcrad}, and a brief presentation of boundary update
+and restart files.
+
+%----------------------------------------------
+\subsection{Boundary geometry}
+\label{OBC_geom}
+%
+First one has to realize that open boundaries may not necessarily be located
+at the extremities of the computational domain. They may exist in the middle
+of the domain, for example at Gibraltar Straits if one wants to avoid including
+the Mediterranean in an Atlantic domain. This flexibility has been found necessary
+for the CLIPPER project \citep{Treguier_al_JGR01}. Because of the complexity of the
+geometry of ocean basins, it may even be necessary to have more than one
+''west'' open boundary, more than one ''north'', etc. This is not possible with
+the OBC option: only one open boundary of each kind, west, east, south and
+north is allowed; these names refer to the grid geometry (not to the direction
+of the geographical ''west'', ''east'', etc).
+
+The open boundary geometry is set by a series of parameters in the module
+\mdl{obc\_par}. For an eastern open boundary, parameters are \jp{lp\_obc\_east}
+(true if an east open boundary exists), \jp{jpieob} the $i$-index along which
+the eastern open boundary (eob) is located, \jp{jpjed} the $j$-index at which
+it starts, and \jp{jpjef} the $j$-index where it ends (note $d$ is for ''d\'{e}but''
+and $f$ for ''fin'' in French). Similar parameters exist for the west, south and
+north cases (Table~\ref{Tab_obc_param}).
+
+
+%--------------------------------------------------TABLE--------------------------------------------------
+\begin{table}[htbp] \begin{center} \begin{tabular}{|l|c|c|c|}
+\hline
+Boundary and & Constant index & Starting index (d\'{e}but) & Ending index (fin) \\
+Logical flag & & & \\
+\hline
+West & \jp{jpiwob} $>= 2$ & \jp{jpjwd}$>= 2$ & \jp{jpjwf}<= \np{jpjglo}-1 \\
+lp\_obc\_west & $i$-index of a $u$ point & $j$ of a $T$ point &$j$ of a $T$ point \\
+\hline
+East & \jp{jpieob}$<=$\np{jpiglo}-2&\jp{jpjed} $>= 2$ & \jp{jpjef}$<=$ \np{jpjglo}-1 \\
+ lp\_obc\_east & $i$-index of a $u$ point & $j$ of a $T$ point & $j$ of a $T$ point \\
+\hline
+South & \jp{jpjsob} $>= 2$ & \jp{jpisd} $>= 2$ & \jp{jpisf}$<=$\np{jpiglo}-1 \\
+lp\_obc\_south & $j$-index of a $v$ point & $i$ of a $T$ point & $i$ of a $T$ point \\
+\hline
+North & \jp{jpjnob} $<=$ \np{jpjglo}-2& \jp{jpind} $>= 2$ & \jp{jpinf}$<=$\np{jpiglo}-1 \\
+lp\_obc\_north & $j$-index of a $v$ point & $i$ of a $T$ point & $i$ of a $T$ point \\
+\hline
+\end{tabular} \end{center}
+\caption{ \label{Tab_obc_param}
+Names of different indices relating to the open boundaries. In the case
+of a completely open ocean domain with four ocean boundaries, the parameters
+take exactly the values indicated.}
+\end{table}
+%------------------------------------------------------------------------------------------------------------
+
+The open boundaries must be along coordinate lines. On the C-grid, the boundary
+itself is along a line of normal velocity points: $v$ points for a zonal open boundary
+(the south or north one), and $u$ points for a meridional open boundary (the west
+or east one). Another constraint is that there still must be a row of masked points
+all around the domain, as if the domain were a closed basin (unless periodic conditions
+are used together with open boundary conditions). Therefore, an open boundary
+cannot be located at the first/last index, namely, 1, \jp{jpiglo} or \jp{jpjglo}. Also,
+the open boundary algorithm involves calculating the normal velocity points situated
+just on the boundary, as well as the tangential velocity and temperature and salinity
+just outside the boundary. This means that for a west/south boundary, normal
+velocities and temperature are calculated at the same index \jp{jpiwob} and
+\jp{jpjsob}, respectively. For an east/north boundary, the normal velocity is
+calculated at index \jp{jpieob} and \jp{jpjnob}, but the ``outside'' temperature is
+at index \jp{jpieob}+1 and \jp{jpjnob}+1. This means that \jp{jpieob}, \jp{jpjnob}
+cannot be bigger than \jp{jpiglo}-2, \jp{jpjglo}-2.
+
+
+The starting and ending indices are to be thought of as $T$ point indices: in many
+cases they indicate the first land $T$-point, at the extremity of an open boundary
+(the coast line follows the $f$ grid points, see Fig.~\ref{Fig_obc_north} for an example
+of a northern open boundary). All indices are relative to the global domain. In the
+free surface case it is possible to have ``ocean corners'', that is, an open boundary
+starting and ending in the ocean.
+
+%>>>>>>>>>>>>>>>>>>>>>>>>>>>>
+\begin{figure}[!t] \begin{center}
+\includegraphics[width=0.70\textwidth]{./TexFiles/Figures/Fig_obc_north.pdf}
+\caption{ \label{Fig_obc_north}
+Localization of the North open boundary points.}
+\end{center} \end{figure}
+%>>>>>>>>>>>>>>>>>>>>>>>>>>>>
+
+Although not compulsory, it is highly recommended that the bathymetry in the
+vicinity of an open boundary follows the following rule: in the direction perpendicular
+to the open line, the water depth should be constant for 4 grid points. This is in
+order to ensure that the radiation condition, which involves model variables next
+to the boundary, is calculated in a consistent way. On Fig.\ref{Fig_obc_north} we
+indicate by an $=$ symbol, the points which should have the same depth. It means
+that at the 4 points near the boundary, the bathymetry is cylindrical \gmcomment{not sure
+why cylindrical}. The line behind the open $T$-line must be 0 in the bathymetry file
+(as shown on Fig.\ref{Fig_obc_north} for example).
+
+%----------------------------------------------
+\subsection{Boundary data}
+\label{OBC_data}
+
+It is necessary to provide information at the boundaries. The simplest case is
+when this information does not change in time and is equal to the initial conditions
+(namelist variable \np{nn\_obcdta}=0). This is the case for the standard configuration
+EEL5 with open boundaries. When (\np{nn\_obcdta}=1), open boundary information
+is read from netcdf files. For convenience the input files are supposed to be similar
+to the ''history'' NEMO output files, for dimension names and variable names.
+Open boundary arrays must be dimensioned according to the parameters of table~
+\ref{Tab_obc_param}: for example, at the western boundary, arrays have a
+dimension of \jp{jpwf}-\jp{jpwd}+1 in the horizontal and \jp{jpk} in the vertical.
+
+When ocean observations are used to generate the boundary data (a hydrographic
+section for example, as in \citet{Treguier_al_JGR01}) it happens often that only the velocity
+normal to the boundary is known, which is the reason why the initial OBC code
+assumes that only $T$, $S$, and the normal velocity ($u$ or $v$) needs to be
+specified. As more and more global model solutions and ocean analysis products
+become available, it will be possible to provide information about all the variables
+(including the tangential velocity) so that the specification of four variables at each
+boundaries will become standard. For the sea surface height, one must distinguish
+between the filtered free surface case and the time-splitting or explicit treatment of
+the free surface.
+ In the first case, it is assumed that the user does not wish to represent high
+ frequency motions such as tides. The boundary condition is thus one of zero
+ normal gradient of sea surface height at the open boundaries, following \citet{Marchesiello2001}.
+No information other than the total velocity needs to be provided at the open
+boundaries in that case. In the other two cases (time splitting or explicit free surface),
+the user must provide barotropic information (sea surface height and barotropic
+velocities) and the use of the Flather algorithm for barotropic variables is
+recommanded. However, this algorithm has not yet been fully tested and bugs
+remain in NEMO v2.3. Users should read the code carefully before using it. Finally,
+in the case of the rigid lid approximation the barotropic streamfunction must be
+provided, as documented in \citet{Treguier_al_JGR01}). This option is no longer
+recommended but remains in NEMO V2.3.
+
+One frequently encountered case is when an open boundary domain is constructed
+from a global or larger scale NEMO configuration. Assuming the domain corresponds
+to indices $ib:ie$, $jb:je$ of the global domain, the bathymetry and forcing of the
+small domain can be created by using the following netcdf utility on the global files:
+ncks -F $-d\;x,ib,ie$ $-d\;y,jb,je$ (part of the nco series of utilities,
+see their \href{http://nco.sourceforge.net}{website}).
+The open boundary files can be constructed using ncks
+commands, following table~\ref{Tab_obc_ind}.
+
+%--------------------------------------------------TABLE--------------------------------------------------
+\begin{table}[htbp] \begin{center} \begin{tabular}{|l|c|c|c|c|c|}
+\hline
+OBC & Variable & file name & Index & Start & end \\
+West & T,S & obcwest\_TS.nc & $ib$+1 & $jb$+1 & $je-1$ \\
+ & U & obcwest\_U.nc & $ib$+1 & $jb$+1 & $je-1$ \\
+ & V & obcwest\_V.nc & $ib$+1 & $jb$+1 & $je-1$ \\
+\hline
+East & T,S & obceast\_TS.nc & $ie$-1 & $jb$+1 & $je-1$ \\
+ & U & obceast\_U.nc & $ie$-2 & $jb$+1 & $je-1$ \\
+ & V & obceast\_V.nc & $ie$-1 & $jb$+1 & $je-1$ \\
+\hline
+South & T,S & obcsouth\_TS.nc & $jb$+1 & $ib$+1 & $ie-1$ \\
+ & U & obcsouth\_U.nc & $jb$+1 & $ib$+1 & $ie-1$ \\
+ & V & obcsouth\_V.nc & $jb$+1 & $ib$+1 & $ie-1$ \\
+\hline
+North & T,S & obcnorth\_TS.nc & $je$-1 & $ib$+1 & $ie-1$ \\
+ & U & obcnorth\_U.nc & $je$-1 & $ib$+1 & $ie-1$ \\
+ & V & obcnorth\_V.nc & $je$-2 & $ib$+1 & $ie-1$ \\
+\hline
+\end{tabular} \end{center}
+\caption{ \label{Tab_obc_ind}
+Requirements for creating open boundary files from a global configuration,
+appropriate for the subdomain of indices $ib:ie$, $jb:je$. ``Index'' designates the
+$i$ or $j$ index along which the $u$ of $v$ boundary point is situated in the global
+configuration, starting and ending with the $j$ or $i$ indices indicated.
+For example, to generate file obcnorth\_V.nc, use the command ncks
+$-F$ $-d\;y,je-2$ $-d\;x,ib+1,ie-1$ }
+\end{table}
+%-----------------------------------------------------------------------------------------------------------
+
+It is assumed that the open boundary files contain the variables for the period of
+the model integration. If the boundary files contain one time frame, the boundary
+data is held fixed in time. If the files contain 12 values, it is assumed that the input
+is a climatology for a repeated annual cycle (corresponding to the case \np{ln\_obc\_clim}
+=true). The case of an arbitrary number of time frames is not yet implemented
+correctly; the user is required to write his own code in the module \mdl{obc\_dta}
+to deal with this situation.
+
+\subsection{Radiation algorithm}
+\label{OBC_rad}
+
+The art of open boundary management consists in applying a constraint strong
+enough that the inner domain "feels" the rest of the ocean, but weak enough
+that perturbations are allowed to leave the domain with minimum false reflections
+of energy. The constraints are specified separately at each boundary as time
+scales for ''inflow'' and ''outflow'' as defined below. The time scales are set (in days)
+by namelist parameters such as \np{rn\_dpein}, \np{rn\_dpeob} for the eastern open
+boundary for example. When both time scales are zero for a given boundary
+($e.g.$ for the western boundary, \jp{lp\_obc\_west}=true, \np{rn\_dpwob}=0 and
+\np{rn\_dpwin}=0) this means that the boundary in question is a ''fixed '' boundary
+where the solution is set exactly by the boundary data. This is not recommended,
+except in combination with increased viscosity in a ''sponge'' layer next to the
+boundary in order to avoid spurious reflections.
+
+
+The radiation\/relaxation \gmcomment{the / doesnt seem to appear in the output}
+algorithm is applied when either relaxation time (for ''inflow'' or ''outflow'') is
+non-zero. It has been developed and tested in the SPEM model and its
+successor ROMS \citep{Barnier1996, Marchesiello2001}, which is an
+$s$-coordinate model on an Arakawa C-grid. Although the algorithm has
+been numerically successful in the CLIPPER Atlantic models, the physics
+do not work as expected \citep{Treguier_al_JGR01}. Users are invited to consider
+open boundary conditions (OBC hereafter) with some scepticism
+\citep{Durran2001, Blayo2005}.
+
+The first part of the algorithm calculates a phase velocity to determine
+whether perturbations tend to propagate toward, or away from, the
+boundary. Let us consider a model variable $\phi$.
+The phase velocities ($C_{\phi x}$,$C_{\phi y}$) for the variable $\phi$,
+in the directions normal and tangential to the boundary are
+\begin{equation} \label{Eq_obc_cphi}
+C_{\phi x} = \frac{ -\phi_{t} }{ ( \phi_{x}^{2} + \phi_{y}^{2}) } \phi_{x}
+\;\;\;\;\; \;\;\;
+C_{\phi y} = \frac{ -\phi_{t} }{ ( \phi_{x}^{2} + \phi_{y}^{2}) } \phi_{y}.
+\end{equation}
+Following \citet{Treguier_al_JGR01} and \citet{Marchesiello2001} we retain only
+the normal component of the velocity, $C_{\phi x}$, setting $C_{\phi y} =0$
+(but unlike the original Orlanski radiation algorithm we retain $\phi_{y}$ in
+the expression for $C_{\phi x}$).
+
+The discrete form of (\ref{Eq_obc_cphi}), described by \citet{Barnier1998},
+takes into account the two rows of grid points situated inside the domain
+next to the boundary, and the three previous time steps ($n$, $n-1$,
+and $n-2$). The same equation can then be discretized at the boundary at
+time steps $n-1$, $n$ and $n+1$ \gmcomment{since the original was three time-level}
+in order to extrapolate for the new boundary value $\phi^{n+1}$.
+
+In the open boundary algorithm as implemented in NEMO v2.3, the new boundary
+values are updated differently depending on the sign of $C_{\phi x}$. Let us take
+an eastern boundary as an example. The solution for variable $\phi$ at the
+boundary is given by a generalized wave equation with phase velocity $C_{\phi}$,
+with the addition of a relaxation term, as:
+\begin{eqnarray}
+\phi_{t} & = & -C_{\phi x} \phi_{x} + \frac{1}{\tau_{o}} (\phi_{c}-\phi)
+ \;\;\; \;\;\; \;\;\; (C_{\phi x} > 0), \label{Eq_obc_rado} \\
+\phi_{t} & = & \frac{1}{\tau_{i}} (\phi_{c}-\phi)
+\;\;\; \;\;\; \;\;\;\;\;\; (C_{\phi x} < 0), \label{Eq_obc_radi}
+\end{eqnarray}
+where $\phi_{c}$ is the estimate of $\phi$ at the boundary, provided as boundary
+data. Note that in (\ref{Eq_obc_rado}), $C_{\phi x}$ is bounded by the ratio
+$\delta x/\delta t$ for stability reasons. When $C_{\phi x}$ is eastward (outward
+propagation), the radiation condition (\ref{Eq_obc_rado}) is used.
+When $C_{\phi x}$ is westward (inward propagation), (\ref{Eq_obc_radi}) is
+used with a strong relaxation to climatology (usually $\tau_{i}=\np{rn\_dpein}=$1~day).
+Equation (\ref{Eq_obc_radi}) is solved with a Euler time-stepping scheme. As a
+consequence, setting $\tau_{i}$ smaller than, or equal to the time step is equivalent
+to a fixed boundary condition. A time scale of one day is usually a good compromise
+which guarantees that the inflow conditions remain close to climatology while ensuring
+numerical stability.
+
+In the case of a western boundary located in the Eastern Atlantic, \citet{Penduff_al_JGR00}
+have been able to implement the radiation algorithm without any boundary data,
+using persistence from the previous time step instead. This solution has not worked
+in other cases \citep{Treguier_al_JGR01}, so that the use of boundary data is recommended.
+Even in the outflow condition (\ref{Eq_obc_rado}), we have found it desirable to
+maintain a weak relaxation to climatology. The time step is usually chosen so as to
+be larger than typical turbulent scales (of order 1000~days \gmcomment{or maybe seconds?}).
+
+The radiation condition is applied to the model variables: temperature, salinity,
+tangential and normal velocities. For normal and tangential velocities, $u$ and $v$,
+radiation is applied with phase velocities calculated from $u$ and $v$ respectively.
+For the radiation of tracers, we use the phase velocity calculated from the tangential
+velocity in order to avoid calculating too many independent radiation velocities and
+because tangential velocities and tracers have the same position along the boundary
+on a C-grid.
+
+\subsection{Domain decomposition (\key{mpp\_mpi})}
+\label{OBC_mpp}
+When \key{mpp\_mpi} is active in the code, the computational domain is divided
+into rectangles that are attributed each to a different processor. The open boundary
+code is ``mpp-compatible'' up to a certain point. The radiation algorithm will not
+work if there is an mpp subdomain boundary parallel to the open boundary at the
+index of the boundary, or the grid point after (outside), or three grid points before
+(inside). On the other hand, there is no problem if an mpp subdomain boundary
+cuts the open boundary perpendicularly. These geometrical limitations must be
+checked for by the user (there is no safeguard in the code).
+The general principle for the open boundary mpp code is that loops over the open
+boundaries {not sure what this means} are performed on local indices (nie0,
+nie1, nje0, nje1 for an eastern boundary for instance) that are initialized in module
+\mdl{obc\_ini}. Those indices have relevant values on the processors that contain
+a segment of an open boundary. For processors that do not include an open
+boundary segment, the indices are such that the calculations within the loops are
+not performed.
+\gmcomment{I dont understand most of the last few sentences}
+
+Arrays of climatological data that are read from files are seen by all processors
+and have the same dimensions for all (for instance, for the eastern boundary,
+uedta(jpjglo,jpk,2)). On the other hand, the arrays for the calculation of radiation
+are local to each processor (uebnd(jpj,jpk,3,3) for instance). This allowed the
+CLIPPER model for example, to save on memory where the eastern boundary
+crossed 8 processors so that \jp{jpj} was much smaller than (\jp{jpjef}-\jp{jpjed}+1).
+
+\subsection{Volume conservation}
+\label{OBC_vol}
+
+It is necessary to control the volume inside a domain when using open boundaries.
+With fixed boundaries, it is enough to ensure that the total inflow/outflow has
+reasonable values (either zero or a value compatible with an observed volume
+balance). When using radiative boundary conditions it is necessary to have a
+volume constraint because each open boundary works independently from the
+others. The methodology used to control this volume is identical to the one
+coded in the ROMS model \citep{Marchesiello2001}.
+
+
+%---------------------------------------- EXTRAS
+\colorbox{yellow}{Explain obc\_vol{\ldots}}
+
+\colorbox{yellow}{OBC algorithm for update, OBC restart, list of routines where obc key appears{\ldots}}
+
+\colorbox{yellow}{OBC rigid lid? {\ldots}}
+
+% ====================================================================
+% Unstructured open boundaries BDY
+% ====================================================================
+\section{Unstructured Open Boundary Conditions (\key{bdy}) (BDY)}
+\label{LBC_bdy}
+
+%-----------------------------------------nambdy--------------------------------------------
+\namdisplay{nambdy}
+%-----------------------------------------------------------------------------------------------
+%-----------------------------------------nambdy_index--------------------------------------------
+\namdisplay{nambdy_index}
+%-----------------------------------------------------------------------------------------------
+%-----------------------------------------nambdy_dta--------------------------------------------
+\namdisplay{nambdy_dta}
+%-----------------------------------------------------------------------------------------------
+%-----------------------------------------nambdy_dta--------------------------------------------
+\namdisplay{nambdy_dta2}
+%-----------------------------------------------------------------------------------------------
+
+Options are defined through the \ngn{nambdy} \ngn{nambdy\_index}
+\ngn{nambdy\_dta} \ngn{nambdy\_dta2} namelist variables.
+The BDY module is an alternative implementation of open boundary
+conditions for regional configurations. It implements the Flow
+Relaxation Scheme algorithm for temperature, salinity, velocities and
+ice fields, and the Flather radiation condition for the depth-mean
+transports. The specification of the location of the open boundary is
+completely flexible and allows for example the open boundary to follow
+an isobath or other irregular contour.
+
+The BDY module was modelled on the OBC module and shares many features
+and a similar coding structure \citep{Chanut2005}.
+
+The BDY module is completely rewritten at NEMO 3.4 and there is a new
+set of namelists. Boundary data files used with earlier versions of
+NEMO may need to be re-ordered to work with this version. See the
+section on the Input Boundary Data Files for details.
+
+%----------------------------------------------
+\subsection{The namelists}
+\label{BDY_namelist}
+
+It is possible to define more than one boundary ``set'' and apply
+different boundary conditions to each set. The number of boundary
+sets is defined by \np{nb\_bdy}. Each boundary set may be defined
+as a set of straight line segments in a namelist
+(\np{ln\_coords\_file}=.false.) or read in from a file
+(\np{ln\_coords\_file}=.true.). If the set is defined in a namelist,
+then the namelists nambdy\_index must be included separately, one for
+each set. If the set is defined by a file, then a
+``coordinates.bdy.nc'' file must be provided. The coordinates.bdy file
+is analagous to the usual NEMO ``coordinates.nc'' file. In the example
+above, there are two boundary sets, the first of which is defined via
+a file and the second is defined in a namelist. For more details of
+the definition of the boundary geometry see section
+\ref{BDY_geometry}.
+
+For each boundary set a boundary
+condition has to be chosen for the barotropic solution (``u2d'':
+sea-surface height and barotropic velocities), for the baroclinic
+velocities (``u3d''), and for the active tracers\footnote{The BDY
+ module does not deal with passive tracers at this version}
+(``tra''). For each set of variables there is a choice of algorithm
+and a choice for the data, eg. for the active tracers the algorithm is
+set by \np{nn\_tra} and the choice of data is set by
+\np{nn\_tra\_dta}.
+
+The choice of algorithm is currently as follows:
+
+\mbox{}
+
+\begin{itemize}
+\item[0.] No boundary condition applied. So the solution will ``see''
+ the land points around the edge of the edge of the domain.
+\item[1.] Flow Relaxation Scheme (FRS) available for all variables.
+\item[2.] Flather radiation scheme for the barotropic variables. The
+ Flather scheme is not compatible with the filtered free surface
+ ({\it dynspg\_ts}).
+\end{itemize}
+
+\mbox{}
+
+The main choice for the boundary data is
+to use initial conditions as boundary data (\np{nn\_tra\_dta}=0) or to
+use external data from a file (\np{nn\_tra\_dta}=1). For the
+barotropic solution there is also the option to use tidal
+harmonic forcing either by itself or in addition to other external
+data.
+
+If external boundary data is required then the nambdy\_dta namelist
+must be defined. One nambdy\_dta namelist is required for each boundary
+set in the order in which the boundary sets are defined in nambdy. In
+the example given, two boundary sets have been defined and so there
+are two nambdy\_dta namelists. The boundary data is read in using the
+fldread module, so the nambdy\_dta namelist is in the format required
+for fldread. For each variable required, the filename, the frequency
+of the files and the frequency of the data in the files is given. Also
+whether or not time-interpolation is required and whether the data is
+climatological (time-cyclic) data. Note that on-the-fly spatial
+interpolation of boundary data is not available at this version.
+
+In the example namelists given, two boundary sets are defined. The
+first set is defined via a file and applies FRS conditions to
+temperature and salinity and Flather conditions to the barotropic
+variables. External data is provided in daily files (from a
+large-scale model). Tidal harmonic forcing is also used. The second
+set is defined in a namelist. FRS conditions are applied on
+temperature and salinity and climatological data is read from external
+files.
+
+%----------------------------------------------
+\subsection{The Flow Relaxation Scheme}
+\label{BDY_FRS_scheme}
+
+The Flow Relaxation Scheme (FRS) \citep{Davies_QJRMS76,Engerdahl_Tel95},
+applies a simple relaxation of the model fields to
+externally-specified values over a zone next to the edge of the model
+domain. Given a model prognostic variable $\Phi$
+\begin{equation} \label{Eq_bdy_frs1}
+\Phi(d) = \alpha(d)\Phi_{e}(d) + (1-\alpha(d))\Phi_{m}(d)\;\;\;\;\; d=1,N
+\end{equation}
+where $\Phi_{m}$ is the model solution and $\Phi_{e}$ is the specified
+external field, $d$ gives the discrete distance from the model
+boundary and $\alpha$ is a parameter that varies from $1$ at $d=1$ to
+a small value at $d=N$. It can be shown that this scheme is equivalent
+to adding a relaxation term to the prognostic equation for $\Phi$ of
+the form:
+\begin{equation} \label{Eq_bdy_frs2}
+-\frac{1}{\tau}\left(\Phi - \Phi_{e}\right)
+\end{equation}
+where the relaxation time scale $\tau$ is given by a function of
+$\alpha$ and the model time step $\Delta t$:
+\begin{equation} \label{Eq_bdy_frs3}
+\tau = \frac{1-\alpha}{\alpha} \,\rdt
+\end{equation}
+Thus the model solution is completely prescribed by the external
+conditions at the edge of the model domain and is relaxed towards the
+external conditions over the rest of the FRS zone. The application of
+a relaxation zone helps to prevent spurious reflection of outgoing
+signals from the model boundary.
+
+The function $\alpha$ is specified as a $tanh$ function:
+\begin{equation} \label{Eq_bdy_frs4}
+\alpha(d) = 1 - \tanh\left(\frac{d-1}{2}\right), \quad d=1,N
+\end{equation}
+The width of the FRS zone is specified in the namelist as
+\np{nn\_rimwidth}. This is typically set to a value between 8 and 10.
+
+%----------------------------------------------
+\subsection{The Flather radiation scheme}
+\label{BDY_flather_scheme}
+
+The \citet{Flather_JPO94} scheme is a radiation condition on the normal, depth-mean
+transport across the open boundary. It takes the form
+\begin{equation} \label{Eq_bdy_fla1}
+U = U_{e} + \frac{c}{h}\left(\eta - \eta_{e}\right),
+\end{equation}
+where $U$ is the depth-mean velocity normal to the boundary and $\eta$
+is the sea surface height, both from the model. The subscript $e$
+indicates the same fields from external sources. The speed of external
+gravity waves is given by $c = \sqrt{gh}$, and $h$ is the depth of the
+water column. The depth-mean normal velocity along the edge of the
+model domain is set equal to the
+external depth-mean normal velocity, plus a correction term that
+allows gravity waves generated internally to exit the model boundary.
+Note that the sea-surface height gradient in \eqref{Eq_bdy_fla1}
+is a spatial gradient across the model boundary, so that $\eta_{e}$ is
+defined on the $T$ points with $nbr=1$ and $\eta$ is defined on the
+$T$ points with $nbr=2$. $U$ and $U_{e}$ are defined on the $U$ or
+$V$ points with $nbr=1$, $i.e.$ between the two $T$ grid points.
+
+%----------------------------------------------
+\subsection{Boundary geometry}
+\label{BDY_geometry}
+
+Each open boundary set is defined as a list of points. The information
+is stored in the arrays $nbi$, $nbj$, and $nbr$ in the $idx\_bdy$
+structure. The $nbi$ and $nbj$ arrays
+define the local $(i,j)$ indices of each point in the boundary zone
+and the $nbr$ array defines the discrete distance from the boundary
+with $nbr=1$ meaning that the point is next to the edge of the
+model domain and $nbr>1$ showing that the point is increasingly
+further away from the edge of the model domain. A set of $nbi$, $nbj$,
+and $nbr$ arrays is defined for each of the $T$, $U$ and $V$
+grids. Figure \ref{Fig_LBC_bdy_geom} shows an example of an irregular
+boundary.
+
+The boundary geometry for each set may be defined in a namelist
+nambdy\_index or by reading in a ``coordinates.bdy.nc'' file. The
+nambdy\_index namelist defines a series of straight-line segments for
+north, east, south and west boundaries. For the northern boundary,
+\np{nbdysegn} gives the number of segments, \np{jpjnob} gives the $j$
+index for each segment and \np{jpindt} and \np{jpinft} give the start
+and end $i$ indices for each segment with similar for the other
+boundaries. These segments define a list of $T$ grid points along the
+outermost row of the boundary ($nbr\,=\, 1$). The code deduces the $U$ and
+$V$ points and also the points for $nbr\,>\, 1$ if
+$nn\_rimwidth\,>\,1$.
+
+The boundary geometry may also be defined from a
+``coordinates.bdy.nc'' file. Figure \ref{Fig_LBC_nc_header}
+gives an example of the header information from such a file. The file
+should contain the index arrays for each of the $T$, $U$ and $V$
+grids. The arrays must be in order of increasing $nbr$. Note that the
+$nbi$, $nbj$ values in the file are global values and are converted to
+local values in the code. Typically this file will be used to generate
+external boundary data via interpolation and so will also contain the
+latitudes and longitudes of each point as shown. However, this is not
+necessary to run the model.
+
+For some choices of irregular boundary the model domain may contain
+areas of ocean which are not part of the computational domain. For
+example if an open boundary is defined along an isobath, say at the
+shelf break, then the areas of ocean outside of this boundary will
+need to be masked out. This can be done by reading a mask file defined
+as \np{cn\_mask\_file} in the nam\_bdy namelist. Only one mask file is
+used even if multiple boundary sets are defined.
+
+%>>>>>>>>>>>>>>>>>>>>>>>>>>>>
+\begin{figure}[!t] \begin{center}
+\includegraphics[width=1.0\textwidth]{./TexFiles/Figures/Fig_LBC_bdy_geom.pdf}
+\caption { \label{Fig_LBC_bdy_geom}
+Example of geometry of unstructured open boundary}
+\end{center} \end{figure}
+%>>>>>>>>>>>>>>>>>>>>>>>>>>>>
+
+%----------------------------------------------
+\subsection{Input boundary data files}
+\label{BDY_data}
+
+The data files contain the data arrays
+in the order in which the points are defined in the $nbi$ and $nbj$
+arrays. The data arrays are dimensioned on: a time dimension;
+$xb$ which is the index of the boundary data point in the horizontal;
+and $yb$ which is a degenerate dimension of 1 to enable the file to be
+read by the standard NEMO I/O routines. The 3D fields also have a
+depth dimension.
+
+At Version 3.4 there are new restrictions on the order in which the
+boundary points are defined (and therefore restrictions on the order
+of the data in the file). In particular:
+
+\mbox{}
+
+\begin{enumerate}
+\item The data points must be in order of increasing $nbr$, ie. all
+ the $nbr=1$ points, then all the $nbr=2$ points etc.
+\item All the data for a particular boundary set must be in the same
+ order. (Prior to 3.4 it was possible to define barotropic data in a
+ different order to the data for tracers and baroclinic velocities).
+\end{enumerate}
+
+\mbox{}
+
+These restrictions mean that data files used with previous versions of
+the model may not work with version 3.4. A fortran utility
+{\it bdy\_reorder} exists in the TOOLS directory which will re-order the
+data in old BDY data files.
+
+%>>>>>>>>>>>>>>>>>>>>>>>>>>>>
+\begin{figure}[!t] \begin{center}
+\includegraphics[width=1.0\textwidth]{./TexFiles/Figures/Fig_LBC_nc_header.pdf}
+\caption { \label{Fig_LBC_nc_header}
+Example of the header for a coordinates.bdy.nc file}
+\end{center} \end{figure}
+%>>>>>>>>>>>>>>>>>>>>>>>>>>>>
+
+%----------------------------------------------
+\subsection{Volume correction}
+\label{BDY_vol_corr}
+
+There is an option to force the total volume in the regional model to be constant,
+similar to the option in the OBC module. This is controlled by the \np{nn\_volctl}
+parameter in the namelist. A value of \np{nn\_volctl}~=~0 indicates that this option is not used.
+If \np{nn\_volctl}~=~1 then a correction is applied to the normal velocities
+around the boundary at each timestep to ensure that the integrated volume flow
+through the boundary is zero. If \np{nn\_volctl}~=~2 then the calculation of
+the volume change on the timestep includes the change due to the freshwater
+flux across the surface and the correction velocity corrects for this as well.
+
+If more than one boundary set is used then volume correction is
+applied to all boundaries at once.
+
+\newpage
+%----------------------------------------------
+\subsection{Tidal harmonic forcing}
+\label{BDY_tides}
+
+%-----------------------------------------nambdy_tide--------------------------------------------
+\namdisplay{nambdy_tide}
+%-----------------------------------------------------------------------------------------------
+
+Options are defined through the \ngn{nambdy\_tide} namelist variables.
+ To be written....
+
+
+
+
Index: /branches/UKMO/dev_r5518_GO6_under_ice_relax_dr_hook/DOC/TexFiles/Chapters/Chap_LDF.tex
===================================================================
--- /branches/UKMO/dev_r5518_GO6_under_ice_relax_dr_hook/DOC/TexFiles/Chapters/Chap_LDF.tex (revision 11643)
+++ /branches/UKMO/dev_r5518_GO6_under_ice_relax_dr_hook/DOC/TexFiles/Chapters/Chap_LDF.tex (revision 11643)
@@ -0,0 +1,473 @@
+
+% ================================================================
+% Chapter � Lateral Ocean Physics (LDF)
+% ================================================================
+\chapter{Lateral Ocean Physics (LDF)}
+\label{LDF}
+\minitoc
+
+
+\newpage
+$\ $\newline % force a new ligne
+
+
+The lateral physics terms in the momentum and tracer equations have been
+described in \S\ref{PE_zdf} and their discrete formulation in \S\ref{TRA_ldf}
+and \S\ref{DYN_ldf}). In this section we further discuss each lateral physics option.
+Choosing one lateral physics scheme means for the user defining, (1) the space
+and time variations of the eddy coefficients ; (2) the direction along which the
+lateral diffusive fluxes are evaluated (model level, geopotential or isopycnal
+surfaces); and (3) the type of operator used (harmonic, or biharmonic operators,
+and for tracers only, eddy induced advection on tracers). These three aspects
+of the lateral diffusion are set through namelist parameters and CPP keys
+(see the \textit{\ngn{nam\_traldf}} and \textit{\ngn{nam\_dynldf}} below). Note
+that this chapter describes the default implementation of iso-neutral
+tracer mixing, and Griffies's implementation, which is used if
+\np{traldf\_grif}=true, is described in Appdx\ref{sec:triad}
+
+%-----------------------------------nam_traldf - nam_dynldf--------------------------------------------
+\namdisplay{namtra_ldf}
+\namdisplay{namdyn_ldf}
+%--------------------------------------------------------------------------------------------------------------
+
+
+% ================================================================
+% Lateral Mixing Coefficients
+% ================================================================
+\section [Lateral Mixing Coefficient (\textit{ldftra}, \textit{ldfdyn})]
+ {Lateral Mixing Coefficient (\mdl{ldftra}, \mdl{ldfdyn}) }
+\label{LDF_coef}
+
+
+Introducing a space variation in the lateral eddy mixing coefficients changes
+the model core memory requirement, adding up to four extra three-dimensional
+arrays for the geopotential or isopycnal second order operator applied to
+momentum. Six CPP keys control the space variation of eddy coefficients:
+three for momentum and three for tracer. The three choices allow:
+a space variation in the three space directions (\key{traldf\_c3d}, \key{dynldf\_c3d}),
+in the horizontal plane (\key{traldf\_c2d}, \key{dynldf\_c2d}),
+or in the vertical only (\key{traldf\_c1d}, \key{dynldf\_c1d}).
+The default option is a constant value over the whole ocean on both momentum and tracers.
+
+The number of additional arrays that have to be defined and the gridpoint
+position at which they are defined depend on both the space variation chosen
+and the type of operator used. The resulting eddy viscosity and diffusivity
+coefficients can be a function of more than one variable. Changes in the
+computer code when switching from one option to another have been
+minimized by introducing the eddy coefficients as statement functions
+(include file \hf{ldftra\_substitute} and \hf{ldfdyn\_substitute}). The functions
+are replaced by their actual meaning during the preprocessing step (CPP).
+The specification of the space variation of the coefficient is made in
+\mdl{ldftra} and \mdl{ldfdyn}, or more precisely in include files
+\textit{traldf\_cNd.h90} and \textit{dynldf\_cNd.h90}, with N=1, 2 or 3.
+The user can modify these include files as he/she wishes. The way the
+mixing coefficient are set in the reference version can be briefly described
+as follows:
+
+\subsubsection{Constant Mixing Coefficients (default option)}
+When none of the \textbf{key\_dynldf\_...} and \textbf{key\_traldf\_...} keys are
+defined, a constant value is used over the whole ocean for momentum and
+tracers, which is specified through the \np{rn\_ahm0} and \np{rn\_aht0} namelist
+parameters.
+
+\subsubsection{Vertically varying Mixing Coefficients (\key{traldf\_c1d} and \key{dynldf\_c1d})}
+The 1D option is only available when using the $z$-coordinate with full step.
+Indeed in all the other types of vertical coordinate, the depth is a 3D function
+of (\textbf{i},\textbf{j},\textbf{k}) and therefore, introducing depth-dependent
+mixing coefficients will require 3D arrays. In the 1D option, a hyperbolic variation
+of the lateral mixing coefficient is introduced in which the surface value is
+\np{rn\_aht0} (\np{rn\_ahm0}), the bottom value is 1/4 of the surface value,
+and the transition takes place around z=300~m with a width of 300~m
+($i.e.$ both the depth and the width of the inflection point are set to 300~m).
+This profile is hard coded in file \hf{traldf\_c1d}, but can be easily modified by users.
+
+\subsubsection{Horizontally Varying Mixing Coefficients (\key{traldf\_c2d} and \key{dynldf\_c2d})}
+By default the horizontal variation of the eddy coefficient depends on the local mesh
+size and the type of operator used:
+\begin{equation} \label{Eq_title}
+ A_l = \left\{
+ \begin{aligned}
+ & \frac{\max(e_1,e_2)}{e_{max}} A_o^l & \text{for laplacian operator } \\
+ & \frac{\max(e_1,e_2)^{3}}{e_{max}^{3}} A_o^l & \text{for bilaplacian operator }
+ \end{aligned} \right.
+\end{equation}
+where $e_{max}$ is the maximum of $e_1$ and $e_2$ taken over the whole masked
+ocean domain, and $A_o^l$ is the \np{rn\_ahm0} (momentum) or \np{rn\_aht0} (tracer)
+namelist parameter. This variation is intended to reflect the lesser need for subgrid
+scale eddy mixing where the grid size is smaller in the domain. It was introduced in
+the context of the DYNAMO modelling project \citep{Willebrand_al_PO01}.
+Note that such a grid scale dependance of mixing coefficients significantly increase
+the range of stability of model configurations presenting large changes in grid pacing
+such as global ocean models. Indeed, in such a case, a constant mixing coefficient
+can lead to a blow up of the model due to large coefficient compare to the smallest
+grid size (see \S\ref{STP_forward_imp}), especially when using a bilaplacian operator.
+
+Other formulations can be introduced by the user for a given configuration.
+For example, in the ORCA2 global ocean model (see Configurations), the laplacian
+viscosity operator uses \np{rn\_ahm0}~= 4.10$^4$ m$^2$/s poleward of 20$^{\circ}$
+north and south and decreases linearly to \np{rn\_aht0}~= 2.10$^3$ m$^2$/s
+at the equator \citep{Madec_al_JPO96, Delecluse_Madec_Bk00}. This modification
+can be found in routine \rou{ldf\_dyn\_c2d\_orca} defined in \mdl{ldfdyn\_c2d}.
+Similar modified horizontal variations can be found with the Antarctic or Arctic
+sub-domain options of ORCA2 and ORCA05 (see \&namcfg namelist).
+
+\subsubsection{Space Varying Mixing Coefficients (\key{traldf\_c3d} and \key{dynldf\_c3d})}
+
+The 3D space variation of the mixing coefficient is simply the combination of the
+1D and 2D cases, $i.e.$ a hyperbolic tangent variation with depth associated with
+a grid size dependence of the magnitude of the coefficient.
+
+\subsubsection{Space and Time Varying Mixing Coefficients}
+
+There is no default specification of space and time varying mixing coefficient.
+The only case available is specific to the ORCA2 and ORCA05 global ocean
+configurations. It provides only a tracer
+mixing coefficient for eddy induced velocity (ORCA2) or both iso-neutral and
+eddy induced velocity (ORCA05) that depends on the local growth rate of
+baroclinic instability. This specification is actually used when an ORCA key
+and both \key{traldf\_eiv} and \key{traldf\_c2d} are defined.
+
+$\ $\newline % force a new ligne
+
+The following points are relevant when the eddy coefficient varies spatially:
+
+(1) the momentum diffusion operator acting along model level surfaces is
+written in terms of curl and divergent components of the horizontal current
+(see \S\ref{PE_ldf}). Although the eddy coefficient could be set to different values
+in these two terms, this option is not currently available.
+
+(2) with an horizontally varying viscosity, the quadratic integral constraints
+on enstrophy and on the square of the horizontal divergence for operators
+acting along model-surfaces are no longer satisfied
+(Appendix~\ref{Apdx_dynldf_properties}).
+
+(3) for isopycnal diffusion on momentum or tracers, an additional purely
+horizontal background diffusion with uniform coefficient can be added by
+setting a non zero value of \np{rn\_ahmb0} or \np{rn\_ahtb0}, a background horizontal
+eddy viscosity or diffusivity coefficient (namelist parameters whose default
+values are $0$). However, the technique used to compute the isopycnal
+slopes is intended to get rid of such a background diffusion, since it introduces
+spurious diapycnal diffusion (see {\S\ref{LDF_slp}).
+
+(4) when an eddy induced advection term is used (\key{traldf\_eiv}), $A^{eiv}$,
+the eddy induced coefficient has to be defined. Its space variations are controlled
+by the same CPP variable as for the eddy diffusivity coefficient ($i.e.$
+\textbf{key\_traldf\_cNd}).
+
+(5) the eddy coefficient associated with a biharmonic operator must be set to a \emph{negative} value.
+
+(6) it is possible to use both the laplacian and biharmonic operators concurrently.
+
+(7) it is possible to run without explicit lateral diffusion on momentum (\np{ln\_dynldf\_lap} =
+\np{ln\_dynldf\_bilap} = false). This is recommended when using the UBS advection
+scheme on momentum (\np{ln\_dynadv\_ubs} = true, see \ref{DYN_adv_ubs})
+and can be useful for testing purposes.
+
+% ================================================================
+% Direction of lateral Mixing
+% ================================================================
+\section [Direction of Lateral Mixing (\textit{ldfslp})]
+ {Direction of Lateral Mixing (\mdl{ldfslp})}
+\label{LDF_slp}
+
+%%%
+\gmcomment{ we should emphasize here that the implementation is a rather old one.
+Better work can be achieved by using \citet{Griffies_al_JPO98, Griffies_Bk04} iso-neutral scheme. }
+
+A direction for lateral mixing has to be defined when the desired operator does
+not act along the model levels. This occurs when $(a)$ horizontal mixing is
+required on tracer or momentum (\np{ln\_traldf\_hor} or \np{ln\_dynldf\_hor})
+in $s$- or mixed $s$-$z$- coordinates, and $(b)$ isoneutral mixing is required
+whatever the vertical coordinate is. This direction of mixing is defined by its
+slopes in the \textbf{i}- and \textbf{j}-directions at the face of the cell of the
+quantity to be diffused. For a tracer, this leads to the following four slopes :
+$r_{1u}$, $r_{1w}$, $r_{2v}$, $r_{2w}$ (see \eqref{Eq_tra_ldf_iso}), while
+for momentum the slopes are $r_{1t}$, $r_{1uw}$, $r_{2f}$, $r_{2uw}$ for
+$u$ and $r_{1f}$, $r_{1vw}$, $r_{2t}$, $r_{2vw}$ for $v$.
+
+%gm% add here afigure of the slope in i-direction
+
+\subsection{slopes for tracer geopotential mixing in the $s$-coordinate}
+
+In $s$-coordinates, geopotential mixing ($i.e.$ horizontal mixing) $r_1$ and
+$r_2$ are the slopes between the geopotential and computational surfaces.
+Their discrete formulation is found by locally solving \eqref{Eq_tra_ldf_iso}
+when the diffusive fluxes in the three directions are set to zero and $T$ is
+assumed to be horizontally uniform, $i.e.$ a linear function of $z_T$, the
+depth of a $T$-point.
+%gm { Steven : My version is obviously wrong since I'm left with an arbitrary constant which is the local vertical temperature gradient}
+
+\begin{equation} \label{Eq_ldfslp_geo}
+\begin{aligned}
+ r_{1u} &= \frac{e_{3u}}{ \left( e_{1u}\;\overline{\overline{e_{3w}}}^{\,i+1/2,\,k} \right)}
+ \;\delta_{i+1/2}[z_t]
+ &\approx \frac{1}{e_{1u}}\; \delta_{i+1/2}[z_t]
+\\
+ r_{2v} &= \frac{e_{3v}}{\left( e_{2v}\;\overline{\overline{e_{3w}}}^{\,j+1/2,\,k} \right)}
+ \;\delta_{j+1/2} [z_t]
+ &\approx \frac{1}{e_{2v}}\; \delta_{j+1/2}[z_t]
+\\
+ r_{1w} &= \frac{1}{e_{1w}}\;\overline{\overline{\delta_{i+1/2}[z_t]}}^{\,i,\,k+1/2}
+ &\approx \frac{1}{e_{1w}}\; \delta_{i+1/2}[z_{uw}]
+ \\
+ r_{2w} &= \frac{1}{e_{2w}}\;\overline{\overline{\delta_{j+1/2}[z_t]}}^{\,j,\,k+1/2}
+ &\approx \frac{1}{e_{2w}}\; \delta_{j+1/2}[z_{vw}]
+ \\
+\end{aligned}
+\end{equation}
+
+%gm% caution I'm not sure the simplification was a good idea!
+
+These slopes are computed once in \rou{ldfslp\_init} when \np{ln\_sco}=True,
+and either \np{ln\_traldf\_hor}=True or \np{ln\_dynldf\_hor}=True.
+
+\subsection{Slopes for tracer iso-neutral mixing}\label{LDF_slp_iso}
+In iso-neutral mixing $r_1$ and $r_2$ are the slopes between the iso-neutral
+and computational surfaces. Their formulation does not depend on the vertical
+coordinate used. Their discrete formulation is found using the fact that the
+diffusive fluxes of locally referenced potential density ($i.e.$ $in situ$ density)
+vanish. So, substituting $T$ by $\rho$ in \eqref{Eq_tra_ldf_iso} and setting the
+diffusive fluxes in the three directions to zero leads to the following definition for
+the neutral slopes:
+
+\begin{equation} \label{Eq_ldfslp_iso}
+\begin{split}
+ r_{1u} &= \frac{e_{3u}}{e_{1u}}\; \frac{\delta_{i+1/2}[\rho]}
+ {\overline{\overline{\delta_{k+1/2}[\rho]}}^{\,i+1/2,\,k}}
+\\
+ r_{2v} &= \frac{e_{3v}}{e_{2v}}\; \frac{\delta_{j+1/2}\left[\rho \right]}
+ {\overline{\overline{\delta_{k+1/2}[\rho]}}^{\,j+1/2,\,k}}
+\\
+ r_{1w} &= \frac{e_{3w}}{e_{1w}}\;
+ \frac{\overline{\overline{\delta_{i+1/2}[\rho]}}^{\,i,\,k+1/2}}
+ {\delta_{k+1/2}[\rho]}
+\\
+ r_{2w} &= \frac{e_{3w}}{e_{2w}}\;
+ \frac{\overline{\overline{\delta_{j+1/2}[\rho]}}^{\,j,\,k+1/2}}
+ {\delta_{k+1/2}[\rho]}
+\\
+\end{split}
+\end{equation}
+
+%gm% rewrite this as the explanation is not very clear !!!
+%In practice, \eqref{Eq_ldfslp_iso} is of little help in evaluating the neutral surface slopes. Indeed, for an unsimplified equation of state, the density has a strong dependancy on pressure (here approximated as the depth), therefore applying \eqref{Eq_ldfslp_iso} using the $in situ$ density, $\rho$, computed at T-points leads to a flattening of slopes as the depth increases. This is due to the strong increase of the $in situ$ density with depth.
+
+%By definition, neutral surfaces are tangent to the local $in situ$ density \citep{McDougall1987}, therefore in \eqref{Eq_ldfslp_iso}, all the derivatives have to be evaluated at the same local pressure (which in decibars is approximated by the depth in meters).
+
+%In the $z$-coordinate, the derivative of the \eqref{Eq_ldfslp_iso} numerator is evaluated at the same depth \nocite{as what?} ($T$-level, which is the same as the $u$- and $v$-levels), so the $in situ$ density can be used for its evaluation.
+
+As the mixing is performed along neutral surfaces, the gradient of $\rho$ in
+\eqref{Eq_ldfslp_iso} has to be evaluated at the same local pressure (which,
+in decibars, is approximated by the depth in meters in the model). Therefore
+\eqref{Eq_ldfslp_iso} cannot be used as such, but further transformation is
+needed depending on the vertical coordinate used:
+
+\begin{description}
+
+\item[$z$-coordinate with full step : ] in \eqref{Eq_ldfslp_iso} the densities
+appearing in the $i$ and $j$ derivatives are taken at the same depth, thus
+the $in situ$ density can be used. This is not the case for the vertical
+derivatives: $\delta_{k+1/2}[\rho]$ is replaced by $-\rho N^2/g$, where $N^2$
+is the local Brunt-Vais\"{a}l\"{a} frequency evaluated following
+\citet{McDougall1987} (see \S\ref{TRA_bn2}).
+
+\item[$z$-coordinate with partial step : ] this case is identical to the full step
+case except that at partial step level, the \emph{horizontal} density gradient
+is evaluated as described in \S\ref{TRA_zpshde}.
+
+\item[$s$- or hybrid $s$-$z$- coordinate : ] in the current release of \NEMO,
+iso-neutral mixing is only employed for $s$-coordinates if the
+Griffies scheme is used (\np{traldf\_grif}=true; see Appdx \ref{sec:triad}).
+In other words, iso-neutral mixing will only be accurately represented with a
+linear equation of state (\np{nn\_eos}=1 or 2). In the case of a "true" equation
+of state, the evaluation of $i$ and $j$ derivatives in \eqref{Eq_ldfslp_iso}
+will include a pressure dependent part, leading to the wrong evaluation of
+the neutral slopes.
+
+%gm%
+Note: The solution for $s$-coordinate passes trough the use of different
+(and better) expression for the constraint on iso-neutral fluxes. Following
+\citet{Griffies_Bk04}, instead of specifying directly that there is a zero neutral
+diffusive flux of locally referenced potential density, we stay in the $T$-$S$
+plane and consider the balance between the neutral direction diffusive fluxes
+of potential temperature and salinity:
+\begin{equation}
+\alpha \ \textbf{F}(T) = \beta \ \textbf{F}(S)
+\end{equation}
+%gm{ where vector F is ....}
+
+This constraint leads to the following definition for the slopes:
+
+\begin{equation} \label{Eq_ldfslp_iso2}
+\begin{split}
+ r_{1u} &= \frac{e_{3u}}{e_{1u}}\; \frac
+ {\alpha_u \;\delta_{i+1/2}[T] - \beta_u \;\delta_{i+1/2}[S]}
+ {\alpha_u \;\overline{\overline{\delta_{k+1/2}[T]}}^{\,i+1/2,\,k}
+ -\beta_u \;\overline{\overline{\delta_{k+1/2}[S]}}^{\,i+1/2,\,k} }
+\\
+ r_{2v} &= \frac{e_{3v}}{e_{2v}}\; \frac
+ {\alpha_v \;\delta_{j+1/2}[T] - \beta_v \;\delta_{j+1/2}[S]}
+ {\alpha_v \;\overline{\overline{\delta_{k+1/2}[T]}}^{\,j+1/2,\,k}
+ -\beta_v \;\overline{\overline{\delta_{k+1/2}[S]}}^{\,j+1/2,\,k} }
+\\
+ r_{1w} &= \frac{e_{3w}}{e_{1w}}\; \frac
+ {\alpha_w \;\overline{\overline{\delta_{i+1/2}[T]}}^{\,i,\,k+1/2}
+ -\beta_w \;\overline{\overline{\delta_{i+1/2}[S]}}^{\,i,\,k+1/2} }
+ {\alpha_w \;\delta_{k+1/2}[T] - \beta_w \;\delta_{k+1/2}[S]}
+\\
+ r_{2w} &= \frac{e_{3w}}{e_{2w}}\; \frac
+ {\alpha_w \;\overline{\overline{\delta_{j+1/2}[T]}}^{\,j,\,k+1/2}
+ -\beta_w \;\overline{\overline{\delta_{j+1/2}[S]}}^{\,j,\,k+1/2} }
+ {\alpha_w \;\delta_{k+1/2}[T] - \beta_w \;\delta_{k+1/2}[S]}
+\\
+\end{split}
+\end{equation}
+where $\alpha$ and $\beta$, the thermal expansion and saline contraction
+coefficients introduced in \S\ref{TRA_bn2}, have to be evaluated at the three
+velocity points. In order to save computation time, they should be approximated
+by the mean of their values at $T$-points (for example in the case of $\alpha$:
+$\alpha_u=\overline{\alpha_T}^{i+1/2}$, $\alpha_v=\overline{\alpha_T}^{j+1/2}$
+and $\alpha_w=\overline{\alpha_T}^{k+1/2}$).
+
+Note that such a formulation could be also used in the $z$-coordinate and
+$z$-coordinate with partial steps cases.
+
+\end{description}
+
+This implementation is a rather old one. It is similar to the one
+proposed by Cox [1987], except for the background horizontal
+diffusion. Indeed, the Cox implementation of isopycnal diffusion in
+GFDL-type models requires a minimum background horizontal diffusion
+for numerical stability reasons. To overcome this problem, several
+techniques have been proposed in which the numerical schemes of the
+ocean model are modified \citep{Weaver_Eby_JPO97,
+ Griffies_al_JPO98}. Griffies's scheme is now available in \NEMO if
+\np{traldf\_grif\_iso} is set true; see Appdx \ref{sec:triad}. Here,
+another strategy is presented \citep{Lazar_PhD97}: a local
+filtering of the iso-neutral slopes (made on 9 grid-points) prevents
+the development of grid point noise generated by the iso-neutral
+diffusion operator (Fig.~\ref{Fig_LDF_ZDF1}). This allows an
+iso-neutral diffusion scheme without additional background horizontal
+mixing. This technique can be viewed as a diffusion operator that acts
+along large-scale (2~$\Delta$x) \gmcomment{2deltax doesnt seem very
+ large scale} iso-neutral surfaces. The diapycnal diffusion required
+for numerical stability is thus minimized and its net effect on the
+flow is quite small when compared to the effect of an horizontal
+background mixing.
+
+Nevertheless, this iso-neutral operator does not ensure that variance cannot increase,
+contrary to the \citet{Griffies_al_JPO98} operator which has that property.
+
+%>>>>>>>>>>>>>>>>>>>>>>>>>>>>
+\begin{figure}[!ht] \begin{center}
+\includegraphics[width=0.70\textwidth]{./TexFiles/Figures/Fig_LDF_ZDF1.pdf}
+\caption { \label{Fig_LDF_ZDF1}
+averaging procedure for isopycnal slope computation.}
+\end{center} \end{figure}
+%>>>>>>>>>>>>>>>>>>>>>>>>>>>>
+
+%There are three additional questions about the slope calculation.
+%First the expression for the rotation tensor has been obtain assuming the "small slope" approximation, so a bound has to be imposed on slopes.
+%Second, numerical stability issues also require a bound on slopes.
+%Third, the question of boundary condition specified on slopes...
+
+%from griffies: chapter 13.1....
+
+
+
+% In addition and also for numerical stability reasons \citep{Cox1987, Griffies_Bk04},
+% the slopes are bounded by $1/100$ everywhere. This limit is decreasing linearly
+% to zero fom $70$ meters depth and the surface (the fact that the eddies "feel" the
+% surface motivates this flattening of isopycnals near the surface).
+
+For numerical stability reasons \citep{Cox1987, Griffies_Bk04}, the slopes must also
+be bounded by $1/100$ everywhere. This constraint is applied in a piecewise linear
+fashion, increasing from zero at the surface to $1/100$ at $70$ metres and thereafter
+decreasing to zero at the bottom of the ocean. (the fact that the eddies "feel" the
+surface motivates this flattening of isopycnals near the surface).
+
+%>>>>>>>>>>>>>>>>>>>>>>>>>>>>
+\begin{figure}[!ht] \begin{center}
+\includegraphics[width=0.70\textwidth]{./TexFiles/Figures/Fig_eiv_slp.pdf}
+\caption { \label{Fig_eiv_slp}
+Vertical profile of the slope used for lateral mixing in the mixed layer :
+\textit{(a)} in the real ocean the slope is the iso-neutral slope in the ocean interior,
+which has to be adjusted at the surface boundary (i.e. it must tend to zero at the
+surface since there is no mixing across the air-sea interface: wall boundary
+condition). Nevertheless, the profile between the surface zero value and the interior
+iso-neutral one is unknown, and especially the value at the base of the mixed layer ;
+\textit{(b)} profile of slope using a linear tapering of the slope near the surface and
+imposing a maximum slope of 1/100 ; \textit{(c)} profile of slope actually used in
+\NEMO: a linear decrease of the slope from zero at the surface to its ocean interior
+value computed just below the mixed layer. Note the huge change in the slope at the
+base of the mixed layer between \textit{(b)} and \textit{(c)}.}
+\end{center} \end{figure}
+%>>>>>>>>>>>>>>>>>>>>>>>>>>>>
+
+\colorbox{yellow}{add here a discussion about the flattening of the slopes, vs tapering the coefficient.}
+
+\subsection{slopes for momentum iso-neutral mixing}
+
+The iso-neutral diffusion operator on momentum is the same as the one used on
+tracers but applied to each component of the velocity separately (see
+\eqref{Eq_dyn_ldf_iso} in section~\ref{DYN_ldf_iso}). The slopes between the
+surface along which the diffusion operator acts and the surface of computation
+($z$- or $s$-surfaces) are defined at $T$-, $f$-, and \textit{uw}- points for the
+$u$-component, and $T$-, $f$- and \textit{vw}- points for the $v$-component.
+They are computed from the slopes used for tracer diffusion, $i.e.$
+\eqref{Eq_ldfslp_geo} and \eqref{Eq_ldfslp_iso} :
+
+\begin{equation} \label{Eq_ldfslp_dyn}
+\begin{aligned}
+&r_{1t}\ \ = \overline{r_{1u}}^{\,i} &&& r_{1f}\ \ &= \overline{r_{1u}}^{\,i+1/2} \\
+&r_{2f} \ \ = \overline{r_{2v}}^{\,j+1/2} &&& r_{2t}\ &= \overline{r_{2v}}^{\,j} \\
+&r_{1uw} = \overline{r_{1w}}^{\,i+1/2} &&\ \ \text{and} \ \ & r_{1vw}&= \overline{r_{1w}}^{\,j+1/2} \\
+&r_{2uw}= \overline{r_{2w}}^{\,j+1/2} &&& r_{2vw}&= \overline{r_{2w}}^{\,j+1/2}\\
+\end{aligned}
+\end{equation}
+
+The major issue remaining is in the specification of the boundary conditions.
+The same boundary conditions are chosen as those used for lateral
+diffusion along model level surfaces, i.e. using the shear computed along
+the model levels and with no additional friction at the ocean bottom (see
+{\S\ref{LBC_coast}).
+
+
+% ================================================================
+% Eddy Induced Mixing
+% ================================================================
+\section [Eddy Induced Velocity (\textit{traadv\_eiv}, \textit{ldfeiv})]
+ {Eddy Induced Velocity (\mdl{traadv\_eiv}, \mdl{ldfeiv})}
+\label{LDF_eiv}
+
+When Gent and McWilliams [1990] diffusion is used (\key{traldf\_eiv} defined),
+an eddy induced tracer advection term is added, the formulation of which
+depends on the slopes of iso-neutral surfaces. Contrary to the case of iso-neutral
+mixing, the slopes used here are referenced to the geopotential surfaces, $i.e.$
+\eqref{Eq_ldfslp_geo} is used in $z$-coordinates, and the sum \eqref{Eq_ldfslp_geo}
++ \eqref{Eq_ldfslp_iso} in $s$-coordinates. The eddy induced velocity is given by:
+\begin{equation} \label{Eq_ldfeiv}
+\begin{split}
+ u^* & = \frac{1}{e_{2u}e_{3u}}\; \delta_k \left[e_{2u} \, A_{uw}^{eiv} \; \overline{r_{1w}}^{\,i+1/2} \right]\\
+v^* & = \frac{1}{e_{1u}e_{3v}}\; \delta_k \left[e_{1v} \, A_{vw}^{eiv} \; \overline{r_{2w}}^{\,j+1/2} \right]\\
+w^* & = \frac{1}{e_{1w}e_{2w}}\; \left\{ \delta_i \left[e_{2u} \, A_{uw}^{eiv} \; \overline{r_{1w}}^{\,i+1/2} \right] + \delta_j \left[e_{1v} \, A_{vw}^{eiv} \; \overline{r_{2w}}^{\,j+1/2} \right] \right\} \\
+\end{split}
+\end{equation}
+where $A^{eiv}$ is the eddy induced velocity coefficient whose value is set
+through \np{rn\_aeiv}, a \textit{nam\_traldf} namelist parameter.
+The three components of the eddy induced velocity are computed and add
+to the eulerian velocity in \mdl{traadv\_eiv}. This has been preferred to a
+separate computation of the advective trends associated with the eiv velocity,
+since it allows us to take advantage of all the advection schemes offered for
+the tracers (see \S\ref{TRA_adv}) and not just the $2^{nd}$ order advection
+scheme as in previous releases of OPA \citep{Madec1998}. This is particularly
+useful for passive tracers where \emph{positivity} of the advection scheme is
+of paramount importance.
+
+At the surface, lateral and bottom boundaries, the eddy induced velocity,
+and thus the advective eddy fluxes of heat and salt, are set to zero.
+
+
+
+
+
Index: /branches/UKMO/dev_r5518_GO6_under_ice_relax_dr_hook/DOC/TexFiles/Chapters/Chap_MISC.tex
===================================================================
--- /branches/UKMO/dev_r5518_GO6_under_ice_relax_dr_hook/DOC/TexFiles/Chapters/Chap_MISC.tex (revision 11643)
+++ /branches/UKMO/dev_r5518_GO6_under_ice_relax_dr_hook/DOC/TexFiles/Chapters/Chap_MISC.tex (revision 11643)
@@ -0,0 +1,639 @@
+% ================================================================
+% Chapter � Miscellaneous Topics
+% ================================================================
+\chapter{Miscellaneous Topics}
+\label{MISC}
+\minitoc
+
+\newpage
+$\ $\newline % force a new ligne
+
+% ================================================================
+% Representation of Unresolved Straits
+% ================================================================
+\section{Representation of Unresolved Straits}
+\label{MISC_strait}
+
+In climate modeling, it often occurs that a crucial connections between water masses
+is broken as the grid mesh is too coarse to resolve narrow straits. For example, coarse
+grid spacing typically closes off the Mediterranean from the Atlantic at the Strait of
+Gibraltar. In this case, it is important for climate models to include the effects of salty
+water entering the Atlantic from the Mediterranean. Likewise, it is important for the
+Mediterranean to replenish its supply of water from the Atlantic to balance the net
+evaporation occurring over the Mediterranean region. This problem occurs even in
+eddy permitting simulations. For example, in ORCA 1/4\deg several straits of the Indonesian
+archipelago (Ombai, Lombok...) are much narrow than even a single ocean grid-point.
+
+We describe briefly here the three methods that can be used in \NEMO to handle
+such improperly resolved straits. The first two consist of opening the strait by hand
+while ensuring that the mass exchanges through the strait are not too large by
+either artificially reducing the surface of the strait grid-cells or, locally increasing
+the lateral friction. In the third one, the strait is closed but exchanges of mass,
+heat and salt across the land are allowed.
+Note that such modifications are so specific to a given configuration that no attempt
+has been made to set them in a generic way. However, examples of how
+they can be set up is given in the ORCA 2\deg and 0.5\deg configurations. For example,
+for details of implementation in ORCA2, search:
+\vspace{-10pt}
+\begin{alltt}
+\tiny
+\begin{verbatim}
+IF( cp_cfg == "orca" .AND. jp_cfg == 2 )
+\end{verbatim}
+\end{alltt}
+
+% -------------------------------------------------------------------------------------------------------------
+% Hand made geometry changes
+% -------------------------------------------------------------------------------------------------------------
+\subsection{Hand made geometry changes}
+\label{MISC_strait_hand}
+
+$\bullet$ reduced scale factor in the cross-strait direction to a value in better agreement
+with the true mean width of the strait. (Fig.~\ref{Fig_MISC_strait_hand}).
+This technique is sometime called "partially open face" or "partially closed cells".
+The key issue here is only to reduce the faces of $T$-cell ($i.e.$ change the value
+of the horizontal scale factors at $u$- or $v$-point) but not the volume of the $T$-cell.
+Indeed, reducing the volume of strait $T$-cell can easily produce a numerical
+instability at that grid point that would require a reduction of the model time step.
+The changes associated with strait management are done in \mdl{domhgr},
+just after the definition or reading of the horizontal scale factors.
+
+$\bullet$ increase of the viscous boundary layer thickness by local increase of the
+fmask value at the coast (Fig.~\ref{Fig_MISC_strait_hand}). This is done in
+\mdl{dommsk} together with the setting of the coastal value of fmask
+(see Section \ref{LBC_coast})
+
+%>>>>>>>>>>>>>>>>>>>>>>>>>>>>
+\begin{figure}[!tbp] \begin{center}
+\includegraphics[width=0.80\textwidth]{./TexFiles/Figures/Fig_Gibraltar.pdf}
+\includegraphics[width=0.80\textwidth]{./TexFiles/Figures/Fig_Gibraltar2.pdf}
+\caption{ \label{Fig_MISC_strait_hand}
+Example of the Gibraltar strait defined in a $1\deg \times 1\deg$ mesh.
+\textit{Top}: using partially open cells. The meridional scale factor at $v$-point
+is reduced on both sides of the strait to account for the real width of the strait
+(about 20 km). Note that the scale factors of the strait $T$-point remains unchanged.
+\textit{Bottom}: using viscous boundary layers. The four fmask parameters
+along the strait coastlines are set to a value larger than 4, $i.e.$ "strong" no-slip
+case (see Fig.\ref{Fig_LBC_shlat}) creating a large viscous boundary layer
+that allows a reduced transport through the strait.}
+\end{center} \end{figure}
+%>>>>>>>>>>>>>>>>>>>>>>>>>>>>
+
+% -------------------------------------------------------------------------------------------------------------
+% Cross Land Advection
+% -------------------------------------------------------------------------------------------------------------
+\subsection{Cross Land Advection (\mdl{tracla})}
+\label{MISC_strait_cla}
+%--------------------------------------------namcla--------------------------------------------------------
+\namdisplay{namcla}
+%--------------------------------------------------------------------------------------------------------------
+
+\colorbox{yellow}{Add a short description of CLA staff here or in lateral boundary condition chapter?}
+Options are defined through the \ngn{namcla} namelist variables.
+
+%The problem is resolved here by allowing the mixing of tracers and mass/volume between non-adjacent water columns at nominated regions within the model. Momentum is not mixed. The scheme conserves total tracer content, and total volume (the latter in $z*$- or $s*$-coordinate), and maintains compatibility between the tracer and mass/volume budgets.
+
+% ================================================================
+% Closed seas
+% ================================================================
+\section{Closed seas (\mdl{closea})}
+\label{MISC_closea}
+
+\colorbox{yellow}{Add here a short description of the way closed seas are managed}
+
+
+% ================================================================
+% Sub-Domain Functionality (\textit{nizoom, njzoom}, namelist parameters)
+% ================================================================
+\section{Sub-Domain Functionality (\np{jpizoom}, \np{jpjzoom})}
+\label{MISC_zoom}
+
+The sub-domain functionality, also improperly called the zoom option
+(improperly because it is not associated with a change in model resolution)
+is a quite simple function that allows a simulation over a sub-domain of an
+already defined configuration ($i.e.$ without defining a new mesh, initial
+state and forcings). This option can be useful for testing the user settings
+of surface boundary conditions, or the initial ocean state of a huge ocean
+model configuration while having a small computer memory requirement.
+It can also be used to easily test specific physics in a sub-domain (for example,
+see \citep{Madec_al_JPO96} for a test of the coupling used in the global ocean
+version of OPA between sea-ice and ocean model over the Arctic or Antarctic
+ocean, using a sub-domain). In the standard model, this option does not
+include any specific treatment for the ocean boundaries of the sub-domain:
+they are considered as artificial vertical walls. Nevertheless, it is quite easy
+to add a restoring term toward a climatology in the vicinity of such boundaries
+(see \S\ref{TRA_dmp}).
+
+In order to easily define a sub-domain over which the computation can be
+performed, the dimension of all input arrays (ocean mesh, bathymetry,
+forcing, initial state, ...) are defined as \np{jpidta}, \np{jpjdta} and \np{jpkdta}
+( in \ngn{namcfg} namelist), while the computational domain is defined through
+\np{jpiglo}, \np{jpjglo} and \jp{jpk} (\ngn{namcfg} namelist). When running the
+model over the whole domain, the user sets \np{jpiglo}=\np{jpidta} \np{jpjglo}=\np{jpjdta}
+and \jp{jpk}=\jp{jpkdta}. When running the model over a sub-domain, the user
+has to provide the size of the sub-domain, (\np{jpiglo}, \np{jpjglo}, \np{jpkglo}),
+and the indices of the south western corner as \np{jpizoom} and \np{jpjzoom} in
+the \ngn{namcfg} namelist (Fig.~\ref{Fig_LBC_zoom}).
+
+Note that a third set of dimensions exist, \jp{jpi}, \jp{jpj} and \jp{jpk} which is
+actually used to perform the computation. It is set by default to \jp{jpi}=\np{jpjglo}
+and \jp{jpj}=\np{jpjglo}, except for massively parallel computing where the
+computational domain is laid out on local processor memories following a 2D
+horizontal splitting. % (see {\S}IV.2-c) ref to the section to be updated
+
+\subsection{Simple subsetting of input files via netCDF attributes}
+
+The extended grids for use with the under-shelf ice cavities will result in redundant rows
+around Antarctica if the ice cavities are not active. A simple mechanism for subsetting
+input files associated with the extended domains has been implemented to avoid the need to
+maintain different sets of input fields for use with or without active ice cavities. The
+existing 'zoom' options are overly complex for this task and marked for deletion anyway.
+This alternative subsetting operates for the j-direction only and works by optionally
+looking for and using a global file attribute (named: \np{open\_ocean\_jstart}) to
+determine the starting j-row for input. The use of this option is best explained with an
+example: Consider an ORCA1 configuration using the extended grid bathymetry and coordinate
+files:
+\vspace{-10pt}
+\begin{alltt}
+\tiny
+\begin{verbatim}
+eORCA1_bathymetry_v2.nc
+eORCA1_coordinates.nc
+\end{verbatim}
+\end{alltt}
+\noindent These files define a horizontal domain of 362x332. Assuming the first row with
+open ocean wet points in the non-isf bathymetry for this set is row 42 (Fortran indexing)
+then the formally correct setting for \np{open\_ocean\_jstart} is 41. Using this value as the
+first row to be read will result in a 362x292 domain which is the same size as the original
+ORCA1 domain. Thus the extended coordinates and bathymetry files can be used with all the
+original input files for ORCA1 if the ice cavities are not active (\np{ln\_isfcav =
+.false.}). Full instructions for achieving this are:
+
+\noindent Add the new attribute to any input files requiring a j-row offset, i.e:
+\vspace{-10pt}
+\begin{alltt}
+\tiny
+\begin{verbatim}
+ncatted -a open_ocean_jstart,global,a,d,41 eORCA1_coordinates.nc
+ncatted -a open_ocean_jstart,global,a,d,41 eORCA1_bathymetry_v2.nc
+\end{verbatim}
+\end{alltt}
+
+\noindent Add the logical switch to \ngn{namcfg} in the configuration namelist and set true:
+%--------------------------------------------namcfg--------------------------------------------------------
+\namdisplay{namcfg_orca1}
+%--------------------------------------------------------------------------------------------------------------
+
+\noindent Note the j-size of the global domain is the (extended j-size minus
+\np{open\_ocean\_jstart} + 1 ) and this must match the size of all datasets other than
+bathymetry and coordinates currently. However the option can be extended to any global, 2D
+and 3D, netcdf, input field by adding the:
+\vspace{-10pt}
+\begin{alltt}
+\tiny
+\begin{verbatim}
+lrowattr=ln_use_jattr
+\end{verbatim}
+\end{alltt}
+optional argument to the appropriate \np{iom\_get} call and the \np{open\_ocean\_jstart} attribute to the corresponding input files. It remains the users responsibility to set \np{jpjdta} and \np{jpjglo} values in the \np{namelist\_cfg} file according to their needs.
+
+%>>>>>>>>>>>>>>>>>>>>>>>>>>>>
+\begin{figure}[!ht] \begin{center}
+\includegraphics[width=0.90\textwidth]{./TexFiles/Figures/Fig_LBC_zoom.pdf}
+\caption{ \label{Fig_LBC_zoom}
+Position of a model domain compared to the data input domain when the zoom functionality is used.}
+\end{center} \end{figure}
+%>>>>>>>>>>>>>>>>>>>>>>>>>>>>
+
+
+% ================================================================
+% Accelerating the Convergence
+% ================================================================
+\section{Accelerating the Convergence (\np{nn\_acc} = 1)}
+\label{MISC_acc}
+%--------------------------------------------namdom-------------------------------------------------------
+\namdisplay{namdom}
+%--------------------------------------------------------------------------------------------------------------
+
+Searching an equilibrium state with an global ocean model requires a very long time
+integration period (a few thousand years for a global model). Due to the size of
+the time step required for numerical stability (less than a few hours),
+this usually requires a large elapsed time. In order to overcome this problem,
+\citet{Bryan1984} introduces a technique that is intended to accelerate
+the spin up to equilibrium. It uses a larger time step in
+the tracer evolution equations than in the momentum evolution
+equations. It does not affect the equilibrium solution but modifies the
+trajectory to reach it.
+
+Options are defined through the \ngn{namdom} namelist variables.
+The acceleration of convergence option is used when \np{nn\_acc}=1. In that case,
+$\rdt=rn\_rdt$ is the time step of dynamics while $\widetilde{\rdt}=rdttra$ is the
+tracer time-step. the former is set from the \np{rn\_rdt} namelist parameter while the latter
+is computed using a hyperbolic tangent profile and the following namelist parameters :
+\np{rn\_rdtmin}, \np{rn\_rdtmax} and \np{rn\_rdth}. Those three parameters correspond
+to the surface value the deep ocean value and the depth at which the transition occurs, respectively.
+The set of prognostic equations to solve becomes:
+\begin{equation} \label{Eq_acc}
+\begin{split}
+\frac{\partial \textbf{U}_h }{\partial t}
+ &\equiv \frac{\textbf{U}_h ^{t+1}-\textbf{U}_h^{t-1} }{2\rdt} = \ldots \\
+\frac{\partial T}{\partial t} &\equiv \frac{T^{t+1}-T^{t-1}}{2 \widetilde{\rdt}} = \ldots \\
+\frac{\partial S}{\partial t} &\equiv \frac{S^{t+1} -S^{t-1}}{2 \widetilde{\rdt}} = \ldots \\
+\end{split}
+\end{equation}
+
+\citet{Bryan1984} has examined the consequences of this distorted physics.
+Free waves have a slower phase speed, their meridional structure is slightly
+modified, and the growth rate of baroclinically unstable waves is reduced
+but with a wider range of instability. This technique is efficient for
+searching for an equilibrium state in coarse resolution models. However its
+application is not suitable for many oceanic problems: it cannot be used for
+transient or time evolving problems (in particular, it is very questionable
+to use this technique when there is a seasonal cycle in the forcing fields),
+and it cannot be used in high-resolution models where baroclinically
+unstable processes are important. Moreover, the vertical variation of
+$\widetilde{ \rdt}$ implies that the heat and salt contents are no longer
+conserved due to the vertical coupling of the ocean level through both
+advection and diffusion. Therefore \np{rn\_rdtmin} = \np{rn\_rdtmax} should be
+a more clever choice.
+
+
+% ================================================================
+% Accuracy and Reproducibility
+% ================================================================
+\section{Accuracy and Reproducibility (\mdl{lib\_fortran})}
+\label{MISC_fortran}
+
+\subsection{Issues with intrinsinc SIGN function (\key{nosignedzero})}
+\label{MISC_sign}
+
+The SIGN(A, B) is the \textsc {Fortran} intrinsic function delivers the magnitude
+of A with the sign of B. For example, SIGN(-3.0,2.0) has the value 3.0.
+The problematic case is when the second argument is zero, because, on platforms
+that support IEEE arithmetic, zero is actually a signed number.
+There is a positive zero and a negative zero.
+
+In \textsc{Fortran}~90, the processor was required always to deliver a positive result for SIGN(A, B)
+if B was zero. Nevertheless, in \textsc{Fortran}~95, the processor is allowed to do the correct thing
+and deliver ABS(A) when B is a positive zero and -ABS(A) when B is a negative zero.
+This change in the specification becomes apparent only when B is of type real, and is zero,
+and the processor is capable of distinguishing between positive and negative zero,
+and B is negative real zero. Then SIGN delivers a negative result where, under \textsc{Fortran}~90
+rules, it used to return a positive result.
+This change may be especially sensitive for the ice model, so we overwrite the intrinsinc
+function with our own function simply performing : \\
+\verb? IF( B >= 0.e0 ) THEN ; SIGN(A,B) = ABS(A) ? \\
+\verb? ELSE ; SIGN(A,B) =-ABS(A) ? \\
+\verb? ENDIF ? \\
+This feature can be found in \mdl{lib\_fortran} module and is effective when \key{nosignedzero}
+is defined. We use a CPP key as the overwritting of a intrinsic function can present
+performance issues with some computers/compilers.
+
+
+\subsection{MPP reproducibility}
+\label{MISC_glosum}
+
+The numerical reproducibility of simulations on distributed memory parallel computers
+is a critical issue. In particular, within NEMO global summation of distributed arrays
+is most susceptible to rounding errors, and their propagation and accumulation cause
+uncertainty in final simulation reproducibility on different numbers of processors.
+To avoid so, based on \citet{He_Ding_JSC01} review of different technics,
+we use a so called self-compensated summation method. The idea is to estimate
+the roundoff error, store it in a buffer, and then add it back in the next addition.
+
+Suppose we need to calculate $b = a_1 + a_2 + a_3$. The following algorithm
+will allow to split the sum in two ($sum_1 = a_{1} + a_{2}$ and $b = sum_2 = sum_1 + a_3$)
+with exactly the same rounding errors as the sum performed all at once.
+\begin{align*}
+ sum_1 \ \ &= a_1 + a_2 \\
+ error_1 &= a_2 + ( a_1 - sum_1 ) \\
+ sum_2 \ \ &= sum_1 + a_3 + error_1 \\
+ error_2 &= a_3 + error_1 + ( sum_1 - sum_2 ) \\
+ b \qquad \ &= sum_2 \\
+\end{align*}
+This feature can be found in \mdl{lib\_fortran} module and is effective when \key{mpp\_rep}.
+In that case, all calls to glob\_sum function (summation over the entire basin excluding
+duplicated rows and columns due to cyclic or north fold boundary condition as well as
+overlap MPP areas).
+Note this implementation may be sensitive to the optimization level.
+
+\subsection{MPP scalability}
+\label{MISC_mppsca}
+
+The default method of communicating values across the north-fold in distributed memory applications
+(\key{mpp\_mpi}) uses a \textsc{MPI\_ALLGATHER} function to exchange values from each processing
+region in the northern row with every other processing region in the northern row. This enables a
+global width array containing the top 4 rows to be collated on every northern row processor and then
+folded with a simple algorithm. Although conceptually simple, this "All to All" communication will
+hamper performance scalability for large numbers of northern row processors. From version 3.4
+onwards an alternative method is available which only performs direct "Peer to Peer" communications
+between each processor and its immediate "neighbours" across the fold line. This is achieved by
+using the default \textsc{MPI\_ALLGATHER} method during initialisation to help identify the "active"
+neighbours. Stored lists of these neighbours are then used in all subsequent north-fold exchanges to
+restrict exchanges to those between associated regions. The collated global width array for each
+region is thus only partially filled but is guaranteed to be set at all the locations actually
+required by each individual for the fold operation. This alternative method should give identical
+results to the default \textsc{ALLGATHER} method and is recommended for large values of \np{jpni}.
+The new method is activated by setting \np{ln\_nnogather} to be true ({\bf nammpp}). The
+reproducibility of results using the two methods should be confirmed for each new, non-reference
+configuration.
+
+% ================================================================
+% Model optimisation, Control Print and Benchmark
+% ================================================================
+\section{Model Optimisation, Control Print and Benchmark}
+\label{MISC_opt}
+%--------------------------------------------namctl-------------------------------------------------------
+\namdisplay{namctl}
+%--------------------------------------------------------------------------------------------------------------
+
+ \gmcomment{why not make these bullets into subsections?}
+Options are defined through the \ngn{namctl} namelist variables.
+
+$\bullet$ Vector optimisation:
+
+\key{vectopt\_loop} enables the internal loops to collapse. This is very
+a very efficient way to increase the length of vector calculations and thus
+to speed up the model on vector computers.
+
+% Add here also one word on NPROMA technique that has been found useless, since compiler have made significant progress during the last decade.
+
+% Add also one word on NEC specific optimisation (Novercheck option for example)
+
+$\bullet$ Control print %: describe here 4 things:
+
+1- \np{ln\_ctl} : compute and print the trends averaged over the interior domain
+in all TRA, DYN, LDF and ZDF modules. This option is very helpful when
+diagnosing the origin of an undesired change in model results.
+
+2- also \np{ln\_ctl} but using the nictl and njctl namelist parameters to check
+the source of differences between mono and multi processor runs.
+
+3- \key{esopa} (to be rename key\_nemo) : which is another option for model
+management. When defined, this key forces the activation of all options and
+CPP keys. For example, all tracer and momentum advection schemes are called!
+Therefore the model results have no physical meaning.
+However, this option forces both the compiler and the model to run through
+all the \textsc{Fortran} lines of the model. This allows the user to check for obvious
+compilation or execution errors with all CPP options, and errors in namelist options.
+
+4- last digit comparison (\np{nn\_bit\_cmp}). In an MPP simulation, the computation of
+a sum over the whole domain is performed as the summation over all processors of
+each of their sums over their interior domains. This double sum never gives exactly
+the same result as a single sum over the whole domain, due to truncation differences.
+The "bit comparison" option has been introduced in order to be able to check that
+mono-processor and multi-processor runs give exactly the same results.
+%THIS is to be updated with the mpp_sum_glo introduced in v3.3
+% nn_bit_cmp today only check that the nn_cla = 0 (no cross land advection)
+
+$\bullet$ Benchmark (\np{nn\_bench}). This option defines a benchmark run based on
+a GYRE configuration (see \S\ref{CFG_gyre}) in which the resolution remains the same
+whatever the domain size. This allows a very large model domain to be used, just by
+changing the domain size (\jp{jpiglo}, \jp{jpjglo}) and without adjusting either the time-step
+or the physical parameterisations.
+
+
+% ================================================================
+% Elliptic solvers (SOL)
+% ================================================================
+\section{Elliptic solvers (SOL)}
+\label{MISC_sol}
+%--------------------------------------------namdom-------------------------------------------------------
+\namdisplay{namsol}
+%--------------------------------------------------------------------------------------------------------------
+
+When the filtered sea surface height option is used, the surface pressure gradient is
+computed in \mdl{dynspg\_flt}. The force added in the momentum equation is solved implicitely.
+It is thus solution of an elliptic equation \eqref{Eq_PE_flt} for which two solvers are available:
+a Successive-Over-Relaxation scheme (SOR) and a preconditioned conjugate gradient
+scheme(PCG) \citep{Madec_al_OM88, Madec_PhD90}. The solver is selected trough the
+the value of \np{nn\_solv} \ngn{namsol} namelist variable.
+
+The PCG is a very efficient method for solving elliptic equations on vector computers.
+It is a fast and rather easy method to use; which are attractive features for a large
+number of ocean situations (variable bottom topography, complex coastal geometry,
+variable grid spacing, open or cyclic boundaries, etc ...). It does not require
+a search for an optimal parameter as in the SOR method. However, the SOR has
+been retained because it is a linear solver, which is a very useful property when
+using the adjoint model of \NEMO.
+
+At each time step, the time derivative of the sea surface height at time step $t+1$
+(or equivalently the divergence of the \textit{after} barotropic transport) that appears
+in the filtering forced is the solution of the elliptic equation obtained from the horizontal
+divergence of the vertical summation of \eqref{Eq_PE_flt}.
+Introducing the following coefficients:
+\begin{equation} \label{Eq_sol_matrix}
+\begin{aligned}
+&c_{i,j}^{NS} &&= {2 \rdt }^2 \; \frac{H_v (i,j) \; e_{1v} (i,j)}{e_{2v}(i,j)} \\
+&c_{i,j}^{EW} &&= {2 \rdt }^2 \; \frac{H_u (i,j) \; e_{2u} (i,j)}{e_{1u}(i,j)} \\
+&b_{i,j} &&= \delta_i \left[ e_{2u}M_u \right] - \delta_j \left[ e_{1v}M_v \right]\ , \\
+\end{aligned}
+\end{equation}
+the resulting five-point finite difference equation is given by:
+\begin{equation} \label{Eq_solmat}
+\begin{split}
+ c_{i+1,j}^{NS} D_{i+1,j} + \; c_{i,j+1}^{EW} D_{i,j+1}
+ + c_{i,j} ^{NS} D_{i-1,j} + \; c_{i,j} ^{EW} D_{i,j-1} & \\
+ - \left(c_{i+1,j}^{NS} + c_{i,j+1}^{EW} + c_{i,j}^{NS} + c_{i,j}^{EW} \right) D_{i,j} &= b_{i,j}
+\end{split}
+\end{equation}
+\eqref{Eq_solmat} is a linear symmetric system of equations. All the elements of
+the corresponding matrix \textbf{A} vanish except those of five diagonals. With
+the natural ordering of the grid points (i.e. from west to east and from
+south to north), the structure of \textbf{A} is block-tridiagonal with
+tridiagonal or diagonal blocks. \textbf{A} is a positive-definite symmetric
+matrix of size $(jpi \cdot jpj)^2$, and \textbf{B}, the right hand side of
+\eqref{Eq_solmat}, is a vector.
+
+Note that in the linear free surface case, the depth that appears in \eqref{Eq_sol_matrix}
+does not vary with time, and thus the matrix can be computed once for all. In non-linear free surface
+(\key{vvl} defined) the matrix have to be updated at each time step.
+
+% -------------------------------------------------------------------------------------------------------------
+% Successive Over Relaxation
+% -------------------------------------------------------------------------------------------------------------
+\subsection{Successive Over Relaxation (\np{nn\_solv}=2, \mdl{solsor})}
+\label{MISC_solsor}
+
+Let us introduce the four cardinal coefficients:
+\begin{align*}
+a_{i,j}^S &= c_{i,j }^{NS}/d_{i,j} &\qquad a_{i,j}^W &= c_{i,j}^{EW}/d_{i,j} \\
+a_{i,j}^E &= c_{i,j+1}^{EW}/d_{i,j} &\qquad a_{i,j}^N &= c_{i+1,j}^{NS}/d_{i,j}
+\end{align*}
+where $d_{i,j} = c_{i,j}^{NS}+ c_{i+1,j}^{NS} + c_{i,j}^{EW} + c_{i,j+1}^{EW}$
+(i.e. the diagonal of the matrix). \eqref{Eq_solmat} can be rewritten as:
+\begin{equation} \label{Eq_solmat_p}
+\begin{split}
+a_{i,j}^{N} D_{i+1,j} +\,a_{i,j}^{E} D_{i,j+1} +\, a_{i,j}^{S} D_{i-1,j} +\,a_{i,j}^{W} D_{i,j-1} - D_{i,j} = \tilde{b}_{i,j}
+\end{split}
+\end{equation}
+with $\tilde b_{i,j} = b_{i,j}/d_{i,j}$. \eqref{Eq_solmat_p} is the equation actually solved
+with the SOR method. This method used is an iterative one. Its algorithm can be
+summarised as follows (see \citet{Haltiner1980} for a further discussion):
+
+initialisation (evaluate a first guess from previous time step computations)
+\begin{equation}
+D_{i,j}^0 = 2 \, D_{i,j}^t - D_{i,j}^{t-1}
+\end{equation}
+iteration $n$, from $n=0$ until convergence, do :
+\begin{equation} \label{Eq_sor_algo}
+\begin{split}
+R_{i,j}^n = &a_{i,j}^{N} D_{i+1,j}^n +\,a_{i,j}^{E} D_{i,j+1} ^n
+ +\, a_{i,j}^{S} D_{i-1,j} ^{n+1}+\,a_{i,j}^{W} D_{i,j-1} ^{n+1}
+ - D_{i,j}^n - \tilde{b}_{i,j} \\
+D_{i,j} ^{n+1} = &D_{i,j} ^{n} + \omega \;R_{i,j}^n
+\end{split}
+\end{equation}
+where \textit{$\omega $ }satisfies $1\leq \omega \leq 2$. An optimal value exists for
+\textit{$\omega$} which significantly accelerates the convergence, but it has to be
+adjusted empirically for each model domain (except for a uniform grid where an
+analytical expression for \textit{$\omega$} can be found \citep{Richtmyer1967}).
+The value of $\omega$ is set using \np{rn\_sor}, a \textbf{namelist} parameter.
+The convergence test is of the form:
+\begin{equation}
+\delta = \frac{\sum\limits_{i,j}{R_{i,j}^n}{R_{i,j}^n}}
+ {\sum\limits_{i,j}{ \tilde{b}_{i,j}^n}{\tilde{b}_{i,j}^n}} \leq \epsilon
+\end{equation}
+where $\epsilon$ is the absolute precision that is required. It is recommended
+that a value smaller or equal to $10^{-6}$ is used for $\epsilon$ since larger
+values may lead to numerically induced basin scale barotropic oscillations.
+The precision is specified by setting \np{rn\_eps} (\textbf{namelist} parameter).
+In addition, two other tests are used to halt the iterative algorithm. They involve
+the number of iterations and the modulus of the right hand side. If the former
+exceeds a specified value, \np{nn\_max} (\textbf{namelist} parameter),
+or the latter is greater than $10^{15}$, the whole model computation is stopped
+and the last computed time step fields are saved in a abort.nc NetCDF file.
+In both cases, this usually indicates that there is something wrong in the model
+configuration (an error in the mesh, the initial state, the input forcing,
+or the magnitude of the time step or of the mixing coefficients). A typical value of
+$nn\_max$ is a few hundred when $\epsilon = 10^{-6}$, increasing to a few
+thousand when $\epsilon = 10^{-12}$.
+The vectorization of the SOR algorithm is not straightforward. The scheme
+contains two linear recurrences on $i$ and $j$. This inhibits the vectorisation.
+\eqref{Eq_sor_algo} can be been rewritten as:
+\begin{equation}
+\begin{split}
+R_{i,j}^n
+= &a_{i,j}^{N} D_{i+1,j}^n +\,a_{i,j}^{E} D_{i,j+1} ^n
+ +\,a_{i,j}^{S} D_{i-1,j} ^{n}+\,_{i,j}^{W} D_{i,j-1} ^{n} - D_{i,j}^n - \tilde{b}_{i,j} \\
+R_{i,j}^n = &R_{i,j}^n - \omega \;a_{i,j}^{S}\; R_{i,j-1}^n \\
+R_{i,j}^n = &R_{i,j}^n - \omega \;a_{i,j}^{W}\; R_{i-1,j}^n
+\end{split}
+\end{equation}
+This technique slightly increases the number of iteration required to reach the convergence,
+but this is largely compensated by the gain obtained by the suppression of the recurrences.
+
+Another technique have been chosen, the so-called red-black SOR. It consist in solving successively
+\eqref{Eq_sor_algo} for odd and even grid points. It also slightly reduced the convergence rate
+but allows the vectorisation. In addition, and this is the reason why it has been chosen, it is able to handle the north fold boundary condition used in ORCA configuration ($i.e.$ tri-polar global ocean mesh).
+
+The SOR method is very flexible and can be used under a wide range of conditions,
+including irregular boundaries, interior boundary points, etc. Proofs of convergence, etc.
+may be found in the standard numerical methods texts for partial differential equations.
+
+% -------------------------------------------------------------------------------------------------------------
+% Preconditioned Conjugate Gradient
+% -------------------------------------------------------------------------------------------------------------
+\subsection{Preconditioned Conjugate Gradient (\np{nn\_solv}=1, \mdl{solpcg}) }
+\label{MISC_solpcg}
+
+\textbf{A} is a definite positive symmetric matrix, thus solving the linear
+system \eqref{Eq_solmat} is equivalent to the minimisation of a quadratic
+functional:
+\begin{equation*}
+\textbf{Ax} = \textbf{b} \leftrightarrow \textbf{x} =\text{inf}_{y} \,\phi (\textbf{y})
+\quad , \qquad
+\phi (\textbf{y}) = 1/2 \langle \textbf{Ay},\textbf{y}\rangle - \langle \textbf{b},\textbf{y} \rangle
+\end{equation*}
+where $\langle , \rangle$ is the canonical dot product. The idea of the
+conjugate gradient method is to search for the solution in the following
+iterative way: assuming that $\textbf{x}^n$ has been obtained, $\textbf{x}^{n+1}$
+is found from $\textbf {x}^{n+1}={\textbf {x}}^n+\alpha^n{\textbf {d}}^n$ which satisfies:
+\begin{equation*}
+{\textbf{ x}}^{n+1}=\text{inf} _{{\textbf{ y}}={\textbf{ x}}^n+\alpha^n \,{\textbf{ d}}^n} \,\phi ({\textbf{ y}})\;\;\Leftrightarrow \;\;\frac{d\phi }{d\alpha}=0
+\end{equation*}
+and expressing $\phi (\textbf{y})$ as a function of \textit{$\alpha $}, we obtain the
+value that minimises the functional:
+\begin{equation*}
+\alpha ^n = \langle{ \textbf{r}^n , \textbf{r}^n} \rangle / \langle {\textbf{ A d}^n, \textbf{d}^n} \rangle
+\end{equation*}
+where $\textbf{r}^n = \textbf{b}-\textbf{A x}^n = \textbf{A} (\textbf{x}-\textbf{x}^n)$
+is the error at rank $n$. The descent vector $\textbf{d}^n$ s chosen to be dependent
+on the error: $\textbf{d}^n = \textbf{r}^n + \beta^n \,\textbf{d}^{n-1}$. $\beta ^n$
+is searched such that the descent vectors form an orthogonal basis for the dot
+product linked to \textbf{A}. Expressing the condition
+$\langle \textbf{A d}^n, \textbf{d}^{n-1} \rangle = 0$ the value of $\beta ^n$ is found:
+ $\beta ^n = \langle{ \textbf{r}^n , \textbf{r}^n} \rangle / \langle {\textbf{r}^{n-1}, \textbf{r}^{n-1}} \rangle$.
+ As a result, the errors $ \textbf{r}^n$ form an orthogonal
+base for the canonic dot product while the descent vectors $\textbf{d}^n$ form
+an orthogonal base for the dot product linked to \textbf{A}. The resulting
+algorithm is thus the following one:
+
+initialisation :
+\begin{equation*}
+\begin{split}
+\textbf{x}^0 &= D_{i,j}^0 = 2 D_{i,j}^t - D_{i,j}^{t-1} \quad, \text{the initial guess } \\
+\textbf{r}^0 &= \textbf{d}^0 = \textbf{b} - \textbf{A x}^0 \\
+\gamma_0 &= \langle{ \textbf{r}^0 , \textbf{r}^0} \rangle
+\end{split}
+\end{equation*}
+
+iteration $n,$ from $n=0$ until convergence, do :
+\begin{equation}
+\begin{split}
+\text{z}^n& = \textbf{A d}^n \\
+\alpha_n &= \gamma_n / \langle{ \textbf{z}^n , \textbf{d}^n} \rangle \\
+\textbf{x}^{n+1} &= \textbf{x}^n + \alpha_n \,\textbf{d}^n \\
+\textbf{r}^{n+1} &= \textbf{r}^n - \alpha_n \,\textbf{z}^n \\
+\gamma_{n+1} &= \langle{ \textbf{r}^{n+1} , \textbf{r}^{n+1}} \rangle \\
+\beta_{n+1} &= \gamma_{n+1}/\gamma_{n} \\
+\textbf{d}^{n+1} &= \textbf{r}^{n+1} + \beta_{n+1}\; \textbf{d}^{n}\\
+\end{split}
+\end{equation}
+
+
+The convergence test is:
+\begin{equation}
+\delta = \gamma_{n}\; / \langle{ \textbf{b} , \textbf{b}} \rangle \leq \epsilon
+\end{equation}
+where $\epsilon $ is the absolute precision that is required. As for the SOR algorithm,
+the whole model computation is stopped when the number of iterations, \np{nn\_max}, or
+the modulus of the right hand side of the convergence equation exceeds a
+specified value (see \S\ref{MISC_solsor} for a further discussion). The required
+precision and the maximum number of iterations allowed are specified by setting
+\np{rn\_eps} and \np{nn\_max} (\textbf{namelist} parameters).
+
+It can be demonstrated that the above algorithm is optimal, provides the exact
+solution in a number of iterations equal to the size of the matrix, and that
+the convergence rate is faster as the matrix is closer to the identity matrix,
+$i.e.$ its eigenvalues are closer to 1. Therefore, it is more efficient to solve
+a better conditioned system which has the same solution. For that purpose,
+we introduce a preconditioning matrix \textbf{Q} which is an approximation
+of \textbf{A} but much easier to invert than \textbf{A}, and solve the system:
+\begin{equation} \label{Eq_pmat}
+\textbf{Q}^{-1} \textbf{A x} = \textbf{Q}^{-1} \textbf{b}
+\end{equation}
+
+The same algorithm can be used to solve \eqref{Eq_pmat} if instead of the
+canonical dot product the following one is used:
+${\langle{ \textbf{a} , \textbf{b}} \rangle}_Q = \langle{ \textbf{a} , \textbf{Q b}} \rangle$, and
+if $\textbf{\~{b}} = \textbf{Q}^{-1}\;\textbf{b}$ and $\textbf{\~{A}} = \textbf{Q}^{-1}\;\textbf{A}$
+are substituted to \textbf{b} and \textbf{A} \citep{Madec_al_OM88}.
+In \NEMO, \textbf{Q} is chosen as the diagonal of \textbf{ A}, i.e. the simplest form for
+\textbf{Q} so that it can be easily inverted. In this case, the discrete formulation of
+\eqref{Eq_pmat} is in fact given by \eqref{Eq_solmat_p} and thus the matrix and
+right hand side are computed independently from the solver used.
+
+% ================================================================
+
+
+
+
+
+
+
+
+
+
+
+
Index: /branches/UKMO/dev_r5518_GO6_under_ice_relax_dr_hook/DOC/TexFiles/Chapters/Chap_Model_Basics.tex
===================================================================
--- /branches/UKMO/dev_r5518_GO6_under_ice_relax_dr_hook/DOC/TexFiles/Chapters/Chap_Model_Basics.tex (revision 11643)
+++ /branches/UKMO/dev_r5518_GO6_under_ice_relax_dr_hook/DOC/TexFiles/Chapters/Chap_Model_Basics.tex (revision 11643)
@@ -0,0 +1,1310 @@
+% ================================================================
+% Chapter 1 Ñ Model Basics
+% ================================================================
+
+\chapter{Model basics}
+\label{PE}
+\minitoc
+
+\newpage
+$\ $\newline % force a new ligne
+
+% ================================================================
+% Primitive Equations
+% ================================================================
+\section{Primitive Equations}
+\label{PE_PE}
+
+% -------------------------------------------------------------------------------------------------------------
+% Vector Invariant Formulation
+% -------------------------------------------------------------------------------------------------------------
+
+\subsection{Vector Invariant Formulation}
+\label{PE_Vector}
+
+
+The ocean is a fluid that can be described to a good approximation by the primitive
+equations, $i.e.$ the Navier-Stokes equations along with a nonlinear equation of
+state which couples the two active tracers (temperature and salinity) to the fluid
+velocity, plus the following additional assumptions made from scale considerations:
+
+\textit{(1) spherical earth approximation: }the geopotential surfaces are assumed to
+be spheres so that gravity (local vertical) is parallel to the earth's radius
+
+\textit{(2) thin-shell approximation: }the ocean depth is neglected compared to the earth's radius
+
+\textit{(3) turbulent closure hypothesis: }the turbulent fluxes (which represent the effect
+of small scale processes on the large-scale) are expressed in terms of large-scale features
+
+\textit{(4) Boussinesq hypothesis:} density variations are neglected except in their
+contribution to the buoyancy force
+
+\textit{(5) Hydrostatic hypothesis: }the vertical momentum equation is reduced to a
+balance between the vertical pressure gradient and the buoyancy force (this removes
+convective processes from the initial Navier-Stokes equations and so convective processes
+must be parameterized instead)
+
+\textit{(6) Incompressibility hypothesis: }the three dimensional divergence of the velocity
+vector is assumed to be zero.
+
+Because the gravitational force is so dominant in the equations of large-scale motions,
+it is useful to choose an orthogonal set of unit vectors (\textbf{i},\textbf{j},\textbf{k}) linked
+to the earth such that \textbf{k} is the local upward vector and (\textbf{i},\textbf{j}) are two
+vectors orthogonal to \textbf{k}, $i.e.$ tangent to the geopotential surfaces. Let us define
+the following variables: \textbf{U} the vector velocity, $\textbf{U}=\textbf{U}_h + w\, \textbf{k}$
+(the subscript $h$ denotes the local horizontal vector, $i.e.$ over the (\textbf{i},\textbf{j}) plane),
+$T$ the potential temperature, $S$ the salinity, \textit{$\rho $} the \textit{in situ} density.
+The vector invariant form of the primitive equations in the (\textbf{i},\textbf{j},\textbf{k})
+vector system provides the following six equations (namely the momentum balance, the
+hydrostatic equilibrium, the incompressibility equation, the heat and salt conservation
+equations and an equation of state):
+\begin{subequations} \label{Eq_PE}
+ \begin{equation} \label{Eq_PE_dyn}
+\frac{\partial {\rm {\bf U}}_h }{\partial t}=
+-\left[ {\left( {\nabla \times {\rm {\bf U}}} \right)\times {\rm {\bf U}}
+ +\frac{1}{2}\nabla \left( {{\rm {\bf U}}^2} \right)} \right]_h
+ -f\;{\rm {\bf k}}\times {\rm {\bf U}}_h
+-\frac{1}{\rho _o }\nabla _h p + {\rm {\bf D}}^{\rm {\bf U}} + {\rm {\bf F}}^{\rm {\bf U}}
+ \end{equation}
+ \begin{equation} \label{Eq_PE_hydrostatic}
+\frac{\partial p }{\partial z} = - \rho \ g
+ \end{equation}
+ \begin{equation} \label{Eq_PE_continuity}
+\nabla \cdot {\bf U}= 0
+ \end{equation}
+\begin{equation} \label{Eq_PE_tra_T}
+\frac{\partial T}{\partial t} = - \nabla \cdot \left( T \ \rm{\bf U} \right) + D^T + F^T
+ \end{equation}
+ \begin{equation} \label{Eq_PE_tra_S}
+\frac{\partial S}{\partial t} = - \nabla \cdot \left( S \ \rm{\bf U} \right) + D^S + F^S
+ \end{equation}
+ \begin{equation} \label{Eq_PE_eos}
+\rho = \rho \left( T,S,p \right)
+ \end{equation}
+\end{subequations}
+where $\nabla$ is the generalised derivative vector operator in $(\bf i,\bf j, \bf k)$ directions,
+$t$ is the time, $z$ is the vertical coordinate, $\rho $ is the \textit{in situ} density given by
+the equation of state (\ref{Eq_PE_eos}), $\rho_o$ is a reference density, $p$ the pressure,
+$f=2 \bf \Omega \cdot \bf k$ is the Coriolis acceleration (where $\bf \Omega$ is the Earth's
+angular velocity vector), and $g$ is the gravitational acceleration.
+${\rm {\bf D}}^{\rm {\bf U}}$, $D^T$ and $D^S$ are the parameterisations of small-scale
+physics for momentum, temperature and salinity, and ${\rm {\bf F}}^{\rm {\bf U}}$, $F^T$
+and $F^S$ surface forcing terms. Their nature and formulation are discussed in
+\S\ref{PE_zdf_ldf} and page \S\ref{PE_boundary_condition}.
+
+.
+
+% -------------------------------------------------------------------------------------------------------------
+% Boundary condition
+% -------------------------------------------------------------------------------------------------------------
+\subsection{Boundary Conditions}
+\label{PE_boundary_condition}
+
+An ocean is bounded by complex coastlines, bottom topography at its base and an air-sea
+or ice-sea interface at its top. These boundaries can be defined by two surfaces, $z=-H(i,j)$
+and $z=\eta(i,j,k,t)$, where $H$ is the depth of the ocean bottom and $\eta$ is the height
+of the sea surface. Both $H$ and $\eta$ are usually referenced to a given surface, $z=0$,
+chosen as a mean sea surface (Fig.~\ref{Fig_ocean_bc}). Through these two boundaries,
+the ocean can exchange fluxes of heat, fresh water, salt, and momentum with the solid earth,
+the continental margins, the sea ice and the atmosphere. However, some of these fluxes are
+so weak that even on climatic time scales of thousands of years they can be neglected.
+In the following, we briefly review the fluxes exchanged at the interfaces between the ocean
+and the other components of the earth system.
+
+%>>>>>>>>>>>>>>>>>>>>>>>>>>>>
+\begin{figure}[!ht] \begin{center}
+\includegraphics[width=0.90\textwidth]{./TexFiles/Figures/Fig_I_ocean_bc.pdf}
+\caption{ \label{Fig_ocean_bc}
+The ocean is bounded by two surfaces, $z=-H(i,j)$ and $z=\eta(i,j,t)$, where $H$
+is the depth of the sea floor and $\eta$ the height of the sea surface.
+Both $H$ and $\eta$ are referenced to $z=0$.}
+\end{center} \end{figure}
+%>>>>>>>>>>>>>>>>>>>>>>>>>>>>
+
+
+\begin{description}
+\item[Land - ocean interface:] the major flux between continental margins and the ocean is
+a mass exchange of fresh water through river runoff. Such an exchange modifies the sea
+surface salinity especially in the vicinity of major river mouths. It can be neglected for short
+range integrations but has to be taken into account for long term integrations as it influences
+the characteristics of water masses formed (especially at high latitudes). It is required in order
+to close the water cycle of the climate system. It is usually specified as a fresh water flux at
+the air-sea interface in the vicinity of river mouths.
+\item[Solid earth - ocean interface:] heat and salt fluxes through the sea floor are small,
+except in special areas of little extent. They are usually neglected in the model
+\footnote{In fact, it has been shown that the heat flux associated with the solid Earth cooling
+($i.e.$the geothermal heating) is not negligible for the thermohaline circulation of the world
+ocean (see \ref{TRA_bbc}).}.
+The boundary condition is thus set to no flux of heat and salt across solid boundaries.
+For momentum, the situation is different. There is no flow across solid boundaries,
+$i.e.$ the velocity normal to the ocean bottom and coastlines is zero (in other words,
+the bottom velocity is parallel to solid boundaries). This kinematic boundary condition
+can be expressed as:
+\begin{equation} \label{Eq_PE_w_bbc}
+w = -{\rm {\bf U}}_h \cdot \nabla _h \left( H \right)
+\end{equation}
+In addition, the ocean exchanges momentum with the earth through frictional processes.
+Such momentum transfer occurs at small scales in a boundary layer. It must be parameterized
+in terms of turbulent fluxes using bottom and/or lateral boundary conditions. Its specification
+depends on the nature of the physical parameterisation used for ${\rm {\bf D}}^{\rm {\bf U}}$
+in \eqref{Eq_PE_dyn}. It is discussed in \S\ref{PE_zdf}, page~\pageref{PE_zdf}.% and Chap. III.6 to 9.
+\item[Atmosphere - ocean interface:] the kinematic surface condition plus the mass flux
+of fresh water PE (the precipitation minus evaporation budget) leads to:
+\begin{equation} \label{Eq_PE_w_sbc}
+w = \frac{\partial \eta }{\partial t}
+ + \left. {{\rm {\bf U}}_h } \right|_{z=\eta } \cdot \nabla _h \left( \eta \right)
+ + P-E
+\end{equation}
+The dynamic boundary condition, neglecting the surface tension (which removes capillary
+waves from the system) leads to the continuity of pressure across the interface $z=\eta$.
+The atmosphere and ocean also exchange horizontal momentum (wind stress), and heat.
+\item[Sea ice - ocean interface:] the ocean and sea ice exchange heat, salt, fresh water
+and momentum. The sea surface temperature is constrained to be at the freezing point
+at the interface. Sea ice salinity is very low ($\sim4-6 \,psu$) compared to those of the
+ocean ($\sim34 \,psu$). The cycle of freezing/melting is associated with fresh water and
+salt fluxes that cannot be neglected.
+\end{description}
+
+
+%\newpage
+%$\ $\newline % force a new ligne
+
+% ================================================================
+% The Horizontal Pressure Gradient
+% ================================================================
+\section{The Horizontal Pressure Gradient }
+\label{PE_hor_pg}
+
+% -------------------------------------------------------------------------------------------------------------
+% Pressure Formulation
+% -------------------------------------------------------------------------------------------------------------
+\subsection{Pressure Formulation}
+\label{PE_p_formulation}
+
+The total pressure at a given depth $z$ is composed of a surface pressure $p_s$ at a
+reference geopotential surface ($z=0$) and a hydrostatic pressure $p_h$ such that:
+$p(i,j,k,t)=p_s(i,j,t)+p_h(i,j,k,t)$. The latter is computed by integrating (\ref{Eq_PE_hydrostatic}),
+assuming that pressure in decibars can be approximated by depth in meters in (\ref{Eq_PE_eos}).
+The hydrostatic pressure is then given by:
+\begin{equation} \label{Eq_PE_pressure}
+p_h \left( {i,j,z,t} \right)
+ = \int_{\varsigma =z}^{\varsigma =0} {g\;\rho \left( {T,S,\varsigma} \right)\;d\varsigma }
+\end{equation}
+ Two strategies can be considered for the surface pressure term: $(a)$ introduce of a
+ new variable $\eta$, the free-surface elevation, for which a prognostic equation can be
+ established and solved; $(b)$ assume that the ocean surface is a rigid lid, on which the
+ pressure (or its horizontal gradient) can be diagnosed. When the former strategy is used,
+ one solution of the free-surface elevation consists of the excitation of external gravity waves.
+ The flow is barotropic and the surface moves up and down with gravity as the restoring force.
+ The phase speed of such waves is high (some hundreds of metres per second) so that
+ the time step would have to be very short if they were present in the model. The latter
+ strategy filters out these waves since the rigid lid approximation implies $\eta=0$, $i.e.$
+ the sea surface is the surface $z=0$. This well known approximation increases the surface
+ wave speed to infinity and modifies certain other longwave dynamics ($e.g.$ barotropic
+ Rossby or planetary waves). The rigid-lid hypothesis is an obsolescent feature in modern
+ OGCMs. It has been available until the release 3.1 of \NEMO, and it has been removed
+ in release 3.2 and followings. Only the free surface formulation is now described in the
+ this document (see the next sub-section).
+
+% -------------------------------------------------------------------------------------------------------------
+% Free Surface Formulation
+% -------------------------------------------------------------------------------------------------------------
+\subsection{Free Surface Formulation}
+\label{PE_free_surface}
+
+In the free surface formulation, a variable $\eta$, the sea-surface height, is introduced
+which describes the shape of the air-sea interface. This variable is solution of a
+prognostic equation which is established by forming the vertical average of the kinematic
+surface condition (\ref{Eq_PE_w_bbc}):
+\begin{equation} \label{Eq_PE_ssh}
+\frac{\partial \eta }{\partial t}=-D+P-E
+ \quad \text{where} \
+D=\nabla \cdot \left[ {\left( {H+\eta } \right) \; {\rm{\bf \overline{U}}}_h \,} \right]
+\end{equation}
+and using (\ref{Eq_PE_hydrostatic}) the surface pressure is given by: $p_s = \rho \, g \, \eta$.
+
+Allowing the air-sea interface to move introduces the external gravity waves (EGWs)
+as a class of solution of the primitive equations. These waves are barotropic because
+of hydrostatic assumption, and their phase speed is quite high. Their time scale is
+short with respect to the other processes described by the primitive equations.
+
+Two choices can be made regarding the implementation of the free surface in the model,
+depending on the physical processes of interest.
+
+$\bullet$ If one is interested in EGWs, in particular the tides and their interaction
+with the baroclinic structure of the ocean (internal waves) possibly in shallow seas,
+then a non linear free surface is the most appropriate. This means that no
+approximation is made in (\ref{Eq_PE_ssh}) and that the variation of the ocean
+volume is fully taken into account. Note that in order to study the fast time scales
+associated with EGWs it is necessary to minimize time filtering effects (use an
+explicit time scheme with very small time step, or a split-explicit scheme with
+reasonably small time step, see \S\ref{DYN_spg_exp} or \S\ref{DYN_spg_ts}.
+
+$\bullet$ If one is not interested in EGW but rather sees them as high frequency
+noise, it is possible to apply an explicit filter to slow down the fastest waves while
+not altering the slow barotropic Rossby waves. If further, an approximative conservation
+of heat and salt contents is sufficient for the problem solved, then it is
+sufficient to solve a linearized version of (\ref{Eq_PE_ssh}), which still allows
+to take into account freshwater fluxes applied at the ocean surface \citep{Roullet_Madec_JGR00}.
+
+The filtering of EGWs in models with a free surface is usually a matter of discretisation
+of the temporal derivatives, using the time splitting method \citep{Killworth_al_JPO91, Zhang_Endoh_JGR92}
+or the implicit scheme \citep{Dukowicz1994}. In \NEMO, we use a slightly different approach
+developed by \citet{Roullet_Madec_JGR00}: the damping of EGWs is ensured by introducing an
+additional force in the momentum equation. \eqref{Eq_PE_dyn} becomes:
+\begin{equation} \label{Eq_PE_flt}
+\frac{\partial {\rm {\bf U}}_h }{\partial t}= {\rm {\bf M}}
+- g \nabla \left( \tilde{\rho} \ \eta \right)
+- g \ T_c \nabla \left( \widetilde{\rho} \ \partial_t \eta \right)
+\end{equation}
+where $T_c$, is a parameter with dimensions of time which characterizes the force,
+$\widetilde{\rho} = \rho / \rho_o$ is the dimensionless density, and $\rm {\bf M}$
+represents the collected contributions of the Coriolis, hydrostatic pressure gradient,
+non-linear and viscous terms in \eqref{Eq_PE_dyn}.
+
+The new force can be interpreted as a diffusion of vertically integrated volume flux divergence.
+The time evolution of $D$ is thus governed by a balance of two terms, $-g$ \textbf{A} $\eta$
+and $g \, T_c \,$ \textbf{A} $D$, associated with a propagative regime and a diffusive regime
+in the temporal spectrum, respectively. In the diffusive regime, the EGWs no longer propagate,
+$i.e.$ they are stationary and damped. The diffusion regime applies to the modes shorter than
+$T_c$. For longer ones, the diffusion term vanishes. Hence, the temporally unresolved EGWs
+can be damped by choosing $T_c > \rdt$. \citet{Roullet_Madec_JGR00} demonstrate that
+(\ref{Eq_PE_flt}) can be integrated with a leap frog scheme except the additional term which
+has to be computed implicitly. This is not surprising since the use of a large time step has a
+necessarily numerical cost. Two gains arise in comparison with the previous formulations.
+Firstly, the damping of EGWs can be quantified through the magnitude of the additional term.
+Secondly, the numerical scheme does not need any tuning. Numerical stability is ensured as
+soon as $T_c > \rdt$.
+
+When the variations of free surface elevation are small compared to the thickness of the first
+model layer, the free surface equation (\ref{Eq_PE_ssh}) can be linearized. As emphasized
+by \citet{Roullet_Madec_JGR00} the linearization of (\ref{Eq_PE_ssh}) has consequences on the
+conservation of salt in the model. With the nonlinear free surface equation, the time evolution
+of the total salt content is
+\begin{equation} \label{Eq_PE_salt_content}
+ \frac{\partial }{\partial t}\int\limits_{D\eta } {S\;dv}
+ =\int\limits_S {S\;(-\frac{\partial \eta }{\partial t}-D+P-E)\;ds}
+\end{equation}
+where $S$ is the salinity, and the total salt is integrated over the whole ocean volume
+$D_\eta$ bounded by the time-dependent free surface. The right hand side (which is an
+integral over the free surface) vanishes when the nonlinear equation (\ref{Eq_PE_ssh})
+is satisfied, so that the salt is perfectly conserved. When the free surface equation is
+linearized, \citet{Roullet_Madec_JGR00} show that the total salt content integrated in the fixed
+volume $D$ (bounded by the surface $z=0$) is no longer conserved:
+\begin{equation} \label{Eq_PE_salt_content_linear}
+ \frac{\partial }{\partial t}\int\limits_D {S\;dv}
+ = - \int\limits_S {S\;\frac{\partial \eta }{\partial t}ds}
+\end{equation}
+
+The right hand side of (\ref{Eq_PE_salt_content_linear}) is small in equilibrium solutions
+\citep{Roullet_Madec_JGR00}. It can be significant when the freshwater forcing is not balanced and
+the globally averaged free surface is drifting. An increase in sea surface height \textit{$\eta $}
+results in a decrease of the salinity in the fixed volume $D$. Even in that case though,
+the total salt integrated in the variable volume $D_{\eta}$ varies much less, since
+(\ref{Eq_PE_salt_content_linear}) can be rewritten as
+\begin{equation} \label{Eq_PE_salt_content_corrected}
+\frac{\partial }{\partial t}\int\limits_{D\eta } {S\;dv}
+=\frac{\partial}{\partial t} \left[ \;{\int\limits_D {S\;dv} +\int\limits_S {S\eta \;ds} } \right]
+=\int\limits_S {\eta \;\frac{\partial S}{\partial t}ds}
+\end{equation}
+
+Although the total salt content is not exactly conserved with the linearized free surface,
+its variations are driven by correlations of the time variation of surface salinity with the
+sea surface height, which is a negligible term. This situation contrasts with the case of
+the rigid lid approximation in which case freshwater forcing is represented by a virtual
+salt flux, leading to a spurious source of salt at the ocean surface
+\citep{Huang_JPO93, Roullet_Madec_JGR00}.
+
+\newpage
+$\ $\newline % force a new ligne
+
+% ================================================================
+% Curvilinear z-coordinate System
+% ================================================================
+\section{Curvilinear \textit{z-}coordinate System}
+\label{PE_zco}
+
+
+% -------------------------------------------------------------------------------------------------------------
+% Tensorial Formalism
+% -------------------------------------------------------------------------------------------------------------
+\subsection{Tensorial Formalism}
+\label{PE_tensorial}
+
+In many ocean circulation problems, the flow field has regions of enhanced dynamics
+($i.e.$ surface layers, western boundary currents, equatorial currents, or ocean fronts).
+The representation of such dynamical processes can be improved by specifically increasing
+the model resolution in these regions. As well, it may be convenient to use a lateral
+boundary-following coordinate system to better represent coastal dynamics. Moreover,
+the common geographical coordinate system has a singular point at the North Pole that
+cannot be easily treated in a global model without filtering. A solution consists of introducing
+an appropriate coordinate transformation that shifts the singular point onto land
+\citep{Madec_Imbard_CD96, Murray_JCP96}. As a consequence, it is important to solve the primitive
+equations in various curvilinear coordinate systems. An efficient way of introducing an
+appropriate coordinate transform can be found when using a tensorial formalism.
+This formalism is suited to any multidimensional curvilinear coordinate system. Ocean
+modellers mainly use three-dimensional orthogonal grids on the sphere (spherical earth
+approximation), with preservation of the local vertical. Here we give the simplified equations
+for this particular case. The general case is detailed by \citet{Eiseman1980} in their survey
+of the conservation laws of fluid dynamics.
+
+Let (\textit{i},\textit{j},\textit{k}) be a set of orthogonal curvilinear coordinates on the sphere
+associated with the positively oriented orthogonal set of unit vectors (\textbf{i},\textbf{j},\textbf{k})
+linked to the earth such that \textbf{k} is the local upward vector and (\textbf{i},\textbf{j}) are
+two vectors orthogonal to \textbf{k}, $i.e.$ along geopotential surfaces (Fig.\ref{Fig_referential}).
+Let $(\lambda,\varphi,z)$ be the geographical coordinate system in which a position is defined
+by the latitude $\varphi(i,j)$, the longitude $\lambda(i,j)$ and the distance from the centre of
+the earth $a+z(k)$ where $a$ is the earth's radius and $z$ the altitude above a reference sea
+level (Fig.\ref{Fig_referential}). The local deformation of the curvilinear coordinate system is
+given by $e_1$, $e_2$ and $e_3$, the three scale factors:
+\begin{equation} \label{Eq_scale_factors}
+\begin{aligned}
+ e_1 &=\left( {a+z} \right)\;\left[ {\left( {\frac{\partial \lambda
+}{\partial i}\cos \varphi } \right)^2+\left( {\frac{\partial \varphi
+}{\partial i}} \right)^2} \right]^{1/2} \\
+ e_2 &=\left( {a+z} \right)\;\left[ {\left( {\frac{\partial \lambda
+}{\partial j}\cos \varphi } \right)^2+\left( {\frac{\partial \varphi
+}{\partial j}} \right)^2} \right]^{1/2} \\
+ e_3 &=\left( {\frac{\partial z}{\partial k}} \right) \\
+ \end{aligned}
+ \end{equation}
+
+%>>>>>>>>>>>>>>>>>>>>>>>>>>>>
+\begin{figure}[!tb] \begin{center}
+\includegraphics[width=0.60\textwidth]{./TexFiles/Figures/Fig_I_earth_referential.pdf}
+\caption{ \label{Fig_referential}
+the geographical coordinate system $(\lambda,\varphi,z)$ and the curvilinear
+coordinate system (\textbf{i},\textbf{j},\textbf{k}). }
+\end{center} \end{figure}
+%>>>>>>>>>>>>>>>>>>>>>>>>>>>>
+
+Since the ocean depth is far smaller than the earth's radius, $a+z$, can be replaced by
+$a$ in (\ref{Eq_scale_factors}) (thin-shell approximation). The resulting horizontal scale
+factors $e_1$, $e_2$ are independent of $k$ while the vertical scale factor is a single
+function of $k$ as \textbf{k} is parallel to \textbf{z}. The scalar and vector operators that
+appear in the primitive equations (Eqs. \eqref{Eq_PE_dyn} to \eqref{Eq_PE_eos}) can
+be written in the tensorial form, invariant in any orthogonal horizontal curvilinear coordinate
+system transformation:
+\begin{subequations} \label{Eq_PE_discrete_operators}
+\begin{equation} \label{Eq_PE_grad}
+\nabla q=\frac{1}{e_1 }\frac{\partial q}{\partial i}\;{\rm {\bf
+i}}+\frac{1}{e_2 }\frac{\partial q}{\partial j}\;{\rm {\bf j}}+\frac{1}{e_3
+}\frac{\partial q}{\partial k}\;{\rm {\bf k}} \\
+\end{equation}
+\begin{equation} \label{Eq_PE_div}
+\nabla \cdot {\rm {\bf A}}
+= \frac{1}{e_1 \; e_2} \left[
+ \frac{\partial \left(e_2 \; a_1\right)}{\partial i }
++\frac{\partial \left(e_1 \; a_2\right)}{\partial j } \right]
++ \frac{1}{e_3} \left[ \frac{\partial a_3}{\partial k } \right]
+\end{equation}
+\begin{equation} \label{Eq_PE_curl}
+ \begin{split}
+\nabla \times \vect{A} =
+ \left[ {\frac{1}{e_2 }\frac{\partial a_3}{\partial j}
+ -\frac{1}{e_3 }\frac{\partial a_2 }{\partial k}} \right] \; \vect{i}
+&+\left[ {\frac{1}{e_3 }\frac{\partial a_1 }{\partial k}
+ -\frac{1}{e_1 }\frac{\partial a_3 }{\partial i}} \right] \; \vect{j} \\
+&+\frac{1}{e_1 e_2 } \left[ {\frac{\partial \left( {e_2 a_2 } \right)}{\partial i}
+ -\frac{\partial \left( {e_1 a_1 } \right)}{\partial j}} \right] \; \vect{k}
+ \end{split}
+\end{equation}
+\begin{equation} \label{Eq_PE_lap}
+\Delta q = \nabla \cdot \left( \nabla q \right)
+\end{equation}
+\begin{equation} \label{Eq_PE_lap_vector}
+\Delta {\rm {\bf A}} =
+ \nabla \left( \nabla \cdot {\rm {\bf A}} \right)
+- \nabla \times \left( \nabla \times {\rm {\bf A}} \right)
+\end{equation}
+\end{subequations}
+where $q$ is a scalar quantity and ${\rm {\bf A}}=(a_1,a_2,a_3)$ a vector in the $(i,j,k)$ coordinate system.
+
+% -------------------------------------------------------------------------------------------------------------
+% Continuous Model Equations
+% -------------------------------------------------------------------------------------------------------------
+\subsection{Continuous Model Equations}
+\label{PE_zco_Eq}
+
+In order to express the Primitive Equations in tensorial formalism, it is necessary to compute
+the horizontal component of the non-linear and viscous terms of the equation using
+\eqref{Eq_PE_grad}) to \eqref{Eq_PE_lap_vector}.
+Let us set $\vect U=(u,v,w)={\vect{U}}_h +w\;\vect{k}$, the velocity in the $(i,j,k)$ coordinate
+system and define the relative vorticity $\zeta$ and the divergence of the horizontal velocity
+field $\chi$, by:
+\begin{equation} \label{Eq_PE_curl_Uh}
+\zeta =\frac{1}{e_1 e_2 }\left[ {\frac{\partial \left( {e_2 \,v}
+\right)}{\partial i}-\frac{\partial \left( {e_1 \,u} \right)}{\partial j}}
+\right]
+\end{equation}
+\begin{equation} \label{Eq_PE_div_Uh}
+\chi =\frac{1}{e_1 e_2 }\left[ {\frac{\partial \left( {e_2 \,u}
+\right)}{\partial i}+\frac{\partial \left( {e_1 \,v} \right)}{\partial j}}
+\right]
+\end{equation}
+
+Using the fact that the horizontal scale factors $e_1$ and $e_2$ are independent of $k$
+and that $e_3$ is a function of the single variable $k$, the nonlinear term of
+\eqref{Eq_PE_dyn} can be transformed as follows:
+\begin{flalign*}
+&\left[ {\left( { \nabla \times {\rm {\bf U}} } \right) \times {\rm {\bf U}}
++\frac{1}{2} \nabla \left( {{\rm {\bf U}}^2} \right)} \right]_h &
+\end{flalign*}
+\begin{flalign*}
+&\qquad=\left( {{\begin{array}{*{20}c}
+ {\left[ { \frac{1}{e_3} \frac{\partial u }{\partial k}
+ -\frac{1}{e_1} \frac{\partial w }{\partial i} } \right] w - \zeta \; v } \\
+ {\zeta \; u - \left[ { \frac{1}{e_2} \frac{\partial w}{\partial j}
+ -\frac{1}{e_3} \frac{\partial v}{\partial k} } \right] \ w} \\
+ \end{array} }} \right)
++\frac{1}{2} \left( {{\begin{array}{*{20}c}
+ { \frac{1}{e_1} \frac{\partial \left( u^2+v^2+w^2 \right)}{\partial i}} \hfill \\
+ { \frac{1}{e_2} \frac{\partial \left( u^2+v^2+w^2 \right)}{\partial j}} \hfill \\
+ \end{array} }} \right) &
+\end{flalign*}
+\begin{flalign*}
+& \qquad =\left( {{ \begin{array}{*{20}c}
+ {-\zeta \; v} \hfill \\
+ { \zeta \; u} \hfill \\
+ \end{array} }} \right)
++\frac{1}{2}\left( {{ \begin{array}{*{20}c}
+ {\frac{1}{e_1 }\frac{\partial \left( {u^2+v^2} \right)}{\partial i}} \hfill \\
+ {\frac{1}{e_2 }\frac{\partial \left( {u^2+v^2} \right)}{\partial j}} \hfill \\
+ \end{array} }} \right)
++\frac{1}{e_3 }\left( {{ \begin{array}{*{20}c}
+ { w \; \frac{\partial u}{\partial k}} \\
+ { w \; \frac{\partial v}{\partial k}} \\
+ \end{array} }} \right)
+-\left( {{ \begin{array}{*{20}c}
+ {\frac{w}{e_1}\frac{\partial w}{\partial i}
+ -\frac{1}{2e_1}\frac{\partial w^2}{\partial i}} \hfill \\
+ {\frac{w}{e_2}\frac{\partial w}{\partial j}
+ -\frac{1}{2e_2}\frac{\partial w^2}{\partial j}} \hfill \\
+ \end{array} }} \right) &
+\end{flalign*}
+
+The last term of the right hand side is obviously zero, and thus the nonlinear term of
+\eqref{Eq_PE_dyn} is written in the $(i,j,k)$ coordinate system:
+\begin{equation} \label{Eq_PE_vector_form}
+\left[ {\left( { \nabla \times {\rm {\bf U}} } \right) \times {\rm {\bf U}}
++\frac{1}{2} \nabla \left( {{\rm {\bf U}}^2} \right)} \right]_h
+=\zeta
+\;{\rm {\bf k}}\times {\rm {\bf U}}_h +\frac{1}{2}\nabla _h \left( {{\rm
+{\bf U}}_h^2 } \right)+\frac{1}{e_3 }w\frac{\partial {\rm {\bf U}}_h
+}{\partial k}
+\end{equation}
+
+This is the so-called \textit{vector invariant form} of the momentum advection term.
+For some purposes, it can be advantageous to write this term in the so-called flux form,
+$i.e.$ to write it as the divergence of fluxes. For example, the first component of
+\eqref{Eq_PE_vector_form} (the $i$-component) is transformed as follows:
+\begin{flalign*}
+&{ \begin{array}{*{20}l}
+\left[ {\left( {\nabla \times \vect{U}} \right)\times \vect{U}
+ +\frac{1}{2}\nabla \left( {\vect{U}}^2 \right)} \right]_i % \\
+%\\
+ = - \zeta \;v
+ + \frac{1}{2\;e_1 } \frac{\partial \left( {u^2+v^2} \right)}{\partial i}
+ + \frac{1}{e_3}w \ \frac{\partial u}{\partial k} \\
+\\
+\qquad =\frac{1}{e_1 \; e_2} \left( -v\frac{\partial \left( {e_2 \,v} \right)}{\partial i}
+ +v\frac{\partial \left( {e_1 \,u} \right)}{\partial j} \right)
++\frac{1}{e_1 e_2 }\left( +e_2 \; u\frac{\partial u}{\partial i}
+ +e_2 \; v\frac{\partial v}{\partial i} \right)
++\frac{1}{e_3} \left( w\;\frac{\partial u}{\partial k} \right) \\
+\end{array} } &
+\end{flalign*}
+\begin{flalign*}
+&{ \begin{array}{*{20}l}
+\qquad =\frac{1}{e_1 \; e_2} \left\{
+ -\left( v^2 \frac{\partial e_2 }{\partial i}
+ +e_2 \,v \frac{\partial v }{\partial i} \right)
++\left( \frac{\partial \left( {e_1 \,u\,v} \right)}{\partial j}
+ -e_1 \,u \frac{\partial v }{\partial j} \right) \right.
+\\ \left. \qquad \qquad \quad
++\left( \frac{\partial \left( {e_2 u\,u} \right)}{\partial i}
+ -u \frac{\partial \left( {e_2 u} \right)}{\partial i} \right)
++e_2 v \frac{\partial v }{\partial i}
+ \right\}
++\frac{1}{e_3} \left(
+ \frac{\partial \left( {w\,u} \right) }{\partial k}
+ -u \frac{\partial w }{\partial k} \right) \\
+\end{array} } &
+\end{flalign*}
+\begin{flalign*}
+&{ \begin{array}{*{20}l}
+\qquad =\frac{1}{e_1 \; e_2} \left(
+ \frac{\partial \left( {e_2 \,u\,u} \right)}{\partial i}
+ + \frac{\partial \left( {e_1 \,u\,v} \right)}{\partial j} \right)
++\frac{1}{e_3 } \frac{\partial \left( {w\,u } \right)}{\partial k}
+\\ \qquad \qquad \quad
++\frac{1}{e_1 e_2 } \left(
+ -u \left( \frac{\partial \left( {e_1 v } \right)}{\partial j}
+ -v\,\frac{\partial e_1 }{\partial j} \right)
+ -u \frac{\partial \left( {e_2 u } \right)}{\partial i}
+ \right)
+ -\frac{1}{e_3 } \frac{\partial w}{\partial k} u
+ +\frac{1}{e_1 e_2 }\left( -v^2\frac{\partial e_2 }{\partial i} \right)
+\end{array} } &
+\end{flalign*}
+\begin{flalign*}
+&{ \begin{array}{*{20}l}
+\qquad = \nabla \cdot \left( {{\rm {\bf U}}\,u} \right)
+- \left( \nabla \cdot {\rm {\bf U}} \right) \ u
++\frac{1}{e_1 e_2 }\left(
+ -v^2 \frac{\partial e_2 }{\partial i}
+ +uv \, \frac{\partial e_1 }{\partial j} \right) \\
+\end{array} } &
+\end{flalign*}
+as $\nabla \cdot {\rm {\bf U}}\;=0$ (incompressibility) it comes:
+\begin{flalign*}
+&{ \begin{array}{*{20}l}
+\qquad = \nabla \cdot \left( {{\rm {\bf U}}\,u} \right)
++ \frac{1}{e_1 e_2 } \left( v \; \frac{\partial e_2}{\partial i}
+ -u \; \frac{\partial e_1}{\partial j} \right) \left( -v \right)
+\end{array} } &
+\end{flalign*}
+
+The flux form of the momentum advection term is therefore given by:
+\begin{multline} \label{Eq_PE_flux_form}
+ \left[
+ \left( {\nabla \times {\rm {\bf U}}} \right) \times {\rm {\bf U}}
++\frac{1}{2} \nabla \left( {{\rm {\bf U}}^2} \right)
+ \right]_h
+\\
+= \nabla \cdot \left( {{\begin{array}{*{20}c} {\rm {\bf U}} \, u \hfill \\
+ {\rm {\bf U}} \, v \hfill \\
+ \end{array} }}
+ \right)
++\frac{1}{e_1 e_2 } \left(
+ v\frac{\partial e_2}{\partial i}
+ -u\frac{\partial e_1}{\partial j}
+ \right) {\rm {\bf k}} \times {\rm {\bf U}}_h
+\end{multline}
+
+The flux form has two terms, the first one is expressed as the divergence of momentum
+fluxes (hence the flux form name given to this formulation) and the second one is due to
+the curvilinear nature of the coordinate system used. The latter is called the \emph{metric}
+term and can be viewed as a modification of the Coriolis parameter:
+\begin{equation} \label{Eq_PE_cor+metric}
+f \to f + \frac{1}{e_1\;e_2} \left( v \frac{\partial e_2}{\partial i}
+ -u \frac{\partial e_1}{\partial j} \right)
+\end{equation}
+
+Note that in the case of geographical coordinate, $i.e.$ when $(i,j) \to (\lambda ,\varphi )$
+and $(e_1 ,e_2) \to (a \,\cos \varphi ,a)$, we recover the commonly used modification of
+the Coriolis parameter $f \to f+(u/a) \tan \varphi$.
+
+
+$\ $\newline % force a new ligne
+
+To sum up, the curvilinear $z$-coordinate equations solved by the ocean model can be
+written in the following tensorial formalism:
+
+\vspace{+10pt}
+$\bullet$ \textbf{Vector invariant form of the momentum equations} :
+
+\begin{subequations} \label{Eq_PE_dyn_vect}
+\begin{equation} \label{Eq_PE_dyn_vect_u} \begin{split}
+\frac{\partial u}{\partial t}
+= + \left( {\zeta +f} \right)\,v
+ - \frac{1}{2\,e_1} \frac{\partial}{\partial i} \left( u^2+v^2 \right)
+ - \frac{1}{e_3 } w \frac{\partial u}{\partial k} & \\
+ - \frac{1}{e_1 } \frac{\partial}{\partial i} \left( \frac{p_s+p_h }{\rho _o} \right)
+ &+ D_u^{\vect{U}} + F_u^{\vect{U}} \\
+\\
+\frac{\partial v}{\partial t} =
+ - \left( {\zeta +f} \right)\,u
+ - \frac{1}{2\,e_2 } \frac{\partial }{\partial j}\left( u^2+v^2 \right)
+ - \frac{1}{e_3 } w \frac{\partial v}{\partial k} & \\
+ - \frac{1}{e_2 } \frac{\partial }{\partial j}\left( \frac{p_s+p_h }{\rho _o} \right)
+ &+ D_v^{\vect{U}} + F_v^{\vect{U}}
+\end{split} \end{equation}
+\end{subequations}
+
+
+\vspace{+10pt}
+$\bullet$ \textbf{flux form of the momentum equations} :
+\begin{subequations} \label{Eq_PE_dyn_flux}
+\begin{multline} \label{Eq_PE_dyn_flux_u}
+\frac{\partial u}{\partial t}=
++ \left( { f + \frac{1}{e_1 \; e_2}
+ \left( v \frac{\partial e_2}{\partial i}
+ -u \frac{\partial e_1}{\partial j} \right)} \right) \, v \\
+- \frac{1}{e_1 \; e_2} \left(
+ \frac{\partial \left( {e_2 \,u\,u} \right)}{\partial i}
+ + \frac{\partial \left( {e_1 \,v\,u} \right)}{\partial j} \right)
+ - \frac{1}{e_3 }\frac{\partial \left( { w\,u} \right)}{\partial k} \\
+- \frac{1}{e_1 }\frac{\partial}{\partial i}\left( \frac{p_s+p_h }{\rho _o} \right)
++ D_u^{\vect{U}} + F_u^{\vect{U}}
+\end{multline}
+\begin{multline} \label{Eq_PE_dyn_flux_v}
+\frac{\partial v}{\partial t}=
+- \left( { f + \frac{1}{e_1 \; e_2}
+ \left( v \frac{\partial e_2}{\partial i}
+ -u \frac{\partial e_1}{\partial j} \right)} \right) \, u \\
+ \frac{1}{e_1 \; e_2} \left(
+ \frac{\partial \left( {e_2 \,u\,v} \right)}{\partial i}
+ + \frac{\partial \left( {e_1 \,v\,v} \right)}{\partial j} \right)
+ - \frac{1}{e_3 } \frac{\partial \left( { w\,v} \right)}{\partial k} \\
+- \frac{1}{e_2 }\frac{\partial }{\partial j}\left( \frac{p_s+p_h }{\rho _o} \right)
++ D_v^{\vect{U}} + F_v^{\vect{U}}
+\end{multline}
+\end{subequations}
+where $\zeta$, the relative vorticity, is given by \eqref{Eq_PE_curl_Uh} and $p_s $,
+the surface pressure, is given by:
+\begin{equation} \label{Eq_PE_spg}
+p_s = \left\{ \begin{split}
+\rho \,g \,\eta & \qquad \qquad \; \qquad \text{ standard free surface} \\
+\rho \,g \,\eta &+ \rho_o \,\mu \,\frac{\partial \eta }{\partial t} \qquad \text{ filtered free surface}
+\end{split}
+\right.
+\end{equation}
+with $\eta$ is solution of \eqref{Eq_PE_ssh}
+
+The vertical velocity and the hydrostatic pressure are diagnosed from the following equations:
+\begin{equation} \label{Eq_w_diag}
+\frac{\partial w}{\partial k}=-\chi \;e_3
+\end{equation}
+\begin{equation} \label{Eq_hp_diag}
+\frac{\partial p_h }{\partial k}=-\rho \;g\;e_3
+\end{equation}
+where the divergence of the horizontal velocity, $\chi$ is given by \eqref{Eq_PE_div_Uh}.
+
+\vspace{+10pt}
+$\bullet$ \textit{tracer equations} :
+\begin{equation} \label{Eq_S}
+\frac{\partial T}{\partial t} =
+-\frac{1}{e_1 e_2 }\left[ { \frac{\partial \left( {e_2 T\,u} \right)}{\partial i}
+ +\frac{\partial \left( {e_1 T\,v} \right)}{\partial j}} \right]
+-\frac{1}{e_3 }\frac{\partial \left( {T\,w} \right)}{\partial k} + D^T + F^T
+\end{equation}
+\begin{equation} \label{Eq_T}
+\frac{\partial S}{\partial t} =
+-\frac{1}{e_1 e_2 }\left[ {\frac{\partial \left( {e_2 S\,u} \right)}{\partial i}
+ +\frac{\partial \left( {e_1 S\,v} \right)}{\partial j}} \right]
+-\frac{1}{e_3 }\frac{\partial \left( {S\,w} \right)}{\partial k} + D^S + F^S
+\end{equation}
+\begin{equation} \label{Eq_rho}
+\rho =\rho \left( {T,S,z(k)} \right)
+\end{equation}
+
+The expression of \textbf{D}$^{U}$, $D^{S}$ and $D^{T}$ depends on the subgrid scale
+parameterisation used. It will be defined in \S\ref{PE_zdf}. The nature and formulation of
+${\rm {\bf F}}^{\rm {\bf U}}$, $F^T$ and $F^S$, the surface forcing terms, are discussed
+in Chapter~\ref{SBC}.
+
+
+\newpage
+$\ $\newline % force a new ligne
+% ================================================================
+% Curvilinear generalised vertical coordinate System
+% ================================================================
+\section{Curvilinear generalised vertical coordinate System}
+\label{PE_gco}
+
+The ocean domain presents a huge diversity of situation in the vertical. First the ocean surface is a time dependent surface (moving surface). Second the ocean floor depends on the geographical position, varying from more than 6,000 meters in abyssal trenches to zero at the coast. Last but not least, the ocean stratification exerts a strong barrier to vertical motions and mixing.
+Therefore, in order to represent the ocean with respect to the first point a space and time dependent vertical coordinate that follows the variation of the sea surface height $e.g.$ an $z$*-coordinate; for the second point, a space variation to fit the change of bottom topography $e.g.$ a terrain-following or $\sigma$-coordinate; and for the third point, one will be tempted to use a space and time dependent coordinate that follows the isopycnal surfaces, $e.g.$ an isopycnic coordinate.
+
+In order to satisfy two or more constrains one can even be tempted to mixed these coordinate systems, as in HYCOM (mixture of $z$-coordinate at the surface, isopycnic coordinate in the ocean interior and $\sigma$ at the ocean bottom) \citep{Chassignet_al_JPO03} or OPA (mixture of $z$-coordinate in vicinity the surface and steep topography areas and $\sigma$-coordinate elsewhere) \citep{Madec_al_JPO96} among others.
+
+In fact one is totally free to choose any space and time vertical coordinate by introducing an arbitrary vertical coordinate :
+\begin{equation} \label{Eq_s}
+s=s(i,j,k,t)
+\end{equation}
+with the restriction that the above equation gives a single-valued monotonic relationship between $s$ and $k$, when $i$, $j$ and $t$ are held fixed. \eqref{Eq_s} is a transformation from the $(i,j,k,t)$ coordinate system with independent variables into the $(i,j,s,t)$ generalised coordinate system with $s$ depending on the other three variables through \eqref{Eq_s}.
+This so-called \textit{generalised vertical coordinate} \citep{Kasahara_MWR74} is in fact an Arbitrary Lagrangian--Eulerian (ALE) coordinate. Indeed, choosing an expression for $s$ is an arbitrary choice that determines which part of the vertical velocity (defined from a fixed referential) will cross the levels (Eulerian part) and which part will be used to move them (Lagrangian part).
+The coordinate is also sometime referenced as an adaptive coordinate \citep{Hofmeister_al_OM09}, since the coordinate system is adapted in the course of the simulation. Its most often used implementation is via an ALE algorithm, in which a pure lagrangian step is followed by regridding and remapping steps, the later step implicitly embedding the vertical advection \citep{Hirt_al_JCP74, Chassignet_al_JPO03, White_al_JCP09}. Here we follow the \citep{Kasahara_MWR74} strategy : a regridding step (an update of the vertical coordinate) followed by an eulerian step with an explicit computation of vertical advection relative to the moving s-surfaces.
+
+%\gmcomment{
+
+%A key point here is that the $s$-coordinate depends on $(i,j)$ ==> horizontal pressure gradient...
+
+the generalized vertical coordinates used in ocean modelling are not orthogonal,
+which contrasts with many other applications in mathematical physics.
+Hence, it is useful to keep in mind the following properties that may seem
+odd on initial encounter.
+
+The horizontal velocity in ocean models measures motions in the horizontal plane,
+perpendicular to the local gravitational field. That is, horizontal velocity is mathematically
+the same regardless the vertical coordinate, be it geopotential, isopycnal, pressure,
+or terrain following. The key motivation for maintaining the same horizontal velocity
+component is that the hydrostatic and geostrophic balances are dominant in the large-scale ocean.
+Use of an alternative quasi-horizontal velocity, for example one oriented parallel
+to the generalized surface, would lead to unacceptable numerical errors.
+Correspondingly, the vertical direction is anti-parallel to the gravitational force in all
+of the coordinate systems. We do not choose the alternative of a quasi-vertical
+direction oriented normal to the surface of a constant generalized vertical coordinate.
+
+It is the method used to measure transport across the generalized vertical coordinate
+surfaces which differs between the vertical coordinate choices. That is, computation
+of the dia-surface velocity component represents the fundamental distinction between
+the various coordinates. In some models, such as geopotential, pressure, and
+terrain following, this transport is typically diagnosed from volume or mass conservation.
+In other models, such as isopycnal layered models, this transport is prescribed based
+on assumptions about the physical processes producing a flux across the layer interfaces.
+
+
+In this section we first establish the PE in the generalised vertical $s$-coordinate,
+then we discuss the particular cases available in \NEMO, namely $z$, $z$*, $s$, and $\tilde z$.
+%}
+
+% -------------------------------------------------------------------------------------------------------------
+% The s-coordinate Formulation
+% -------------------------------------------------------------------------------------------------------------
+\subsection{The \textit{s-}coordinate Formulation}
+
+Starting from the set of equations established in \S\ref{PE_zco} for the special case $k=z$
+and thus $e_3=1$, we introduce an arbitrary vertical coordinate $s=s(i,j,k,t)$, which includes
+$z$-, \textit{z*}- and $\sigma-$coordinates as special cases ($s=z$, $s=\textit{z*}$, and
+$s=\sigma=z/H$ or $=z/\left(H+\eta \right)$, resp.). A formal derivation of the transformed
+equations is given in Appendix~\ref{Apdx_A}. Let us define the vertical scale factor by
+$e_3=\partial_s z$ ($e_3$ is now a function of $(i,j,k,t)$ ), and the slopes in the
+(\textbf{i},\textbf{j}) directions between $s-$ and $z-$surfaces by :
+\begin{equation} \label{Eq_PE_sco_slope}
+\sigma _1 =\frac{1}{e_1 }\;\left. {\frac{\partial z}{\partial i}} \right|_s
+\quad \text{, and } \quad
+\sigma _2 =\frac{1}{e_2 }\;\left. {\frac{\partial z}{\partial j}} \right|_s
+\end{equation}
+We also introduce $\omega $, a dia-surface velocity component, defined as the velocity
+relative to the moving $s$-surfaces and normal to them:
+\begin{equation} \label{Eq_PE_sco_w}
+\omega = w - e_3 \, \frac{\partial s}{\partial t} - \sigma _1 \,u - \sigma _2 \,v \\
+\end{equation}
+
+The equations solved by the ocean model \eqref{Eq_PE} in $s-$coordinate can be written as follows:
+
+ \vspace{0.5cm}
+* momentum equation:
+\begin{multline} \label{Eq_PE_sco_u}
+\frac{1}{e_3} \frac{\partial \left( e_3\,u \right) }{\partial t}=
+ + \left( {\zeta +f} \right)\,v
+ - \frac{1}{2\,e_1} \frac{\partial}{\partial i} \left( u^2+v^2 \right)
+ - \frac{1}{e_3} \omega \frac{\partial u}{\partial k} \\
+ - \frac{1}{e_1} \frac{\partial}{\partial i} \left( \frac{p_s + p_h}{\rho _o} \right)
+ + g\frac{\rho }{\rho _o}\sigma _1
+ + D_u^{\vect{U}} + F_u^{\vect{U}} \quad
+\end{multline}
+\begin{multline} \label{Eq_PE_sco_v}
+\frac{1}{e_3} \frac{\partial \left( e_3\,v \right) }{\partial t}=
+ - \left( {\zeta +f} \right)\,u
+ - \frac{1}{2\,e_2 }\frac{\partial }{\partial j}\left( u^2+v^2 \right)
+ - \frac{1}{e_3 } \omega \frac{\partial v}{\partial k} \\
+ - \frac{1}{e_2 }\frac{\partial }{\partial j}\left( \frac{p_s+p_h }{\rho _o} \right)
+ + g\frac{\rho }{\rho _o }\sigma _2
+ + D_v^{\vect{U}} + F_v^{\vect{U}} \quad
+\end{multline}
+where the relative vorticity, \textit{$\zeta $}, the surface pressure gradient, and the hydrostatic
+pressure have the same expressions as in $z$-coordinates although they do not represent
+exactly the same quantities. $\omega$ is provided by the continuity equation
+(see Appendix~\ref{Apdx_A}):
+
+\begin{equation} \label{Eq_PE_sco_continuity}
+\frac{\partial e_3}{\partial t} + e_3 \; \chi + \frac{\partial \omega }{\partial s} = 0
+\qquad \text{with }\;\;
+\chi =\frac{1}{e_1 e_2 e_3 }\left[ {\frac{\partial \left( {e_2 e_3 \,u}
+\right)}{\partial i}+\frac{\partial \left( {e_1 e_3 \,v} \right)}{\partial
+j}} \right]
+\end{equation}
+
+ \vspace{0.5cm}
+* tracer equations:
+\begin{multline} \label{Eq_PE_sco_t}
+\frac{1}{e_3} \frac{\partial \left( e_3\,T \right) }{\partial t}=
+-\frac{1}{e_1 e_2 e_3 }\left[ {\frac{\partial \left( {e_2 e_3\,u\,T} \right)}{\partial i}
+ +\frac{\partial \left( {e_1 e_3\,v\,T} \right)}{\partial j}} \right] \\
+-\frac{1}{e_3 }\frac{\partial \left( {T\,\omega } \right)}{\partial k} + D^T + F^S \qquad
+\end{multline}
+
+\begin{multline} \label{Eq_PE_sco_s}
+\frac{1}{e_3} \frac{\partial \left( e_3\,S \right) }{\partial t}=
+-\frac{1}{e_1 e_2 e_3 }\left[ {\frac{\partial \left( {e_2 e_3\,u\,S} \right)}{\partial i}
+ +\frac{\partial \left( {e_1 e_3\,v\,S} \right)}{\partial j}} \right] \\
+-\frac{1}{e_3 }\frac{\partial \left( {S\,\omega } \right)}{\partial k} + D^S + F^S \qquad
+\end{multline}
+
+The equation of state has the same expression as in $z$-coordinate, and similar expressions
+are used for mixing and forcing terms.
+
+\gmcomment{
+\colorbox{yellow}{ to be updated $= = >$}
+Add a few works on z and zps and s and underlies the differences between all of them
+\colorbox{yellow}{ $< = =$ end update} }
+
+
+
+% -------------------------------------------------------------------------------------------------------------
+% Curvilinear z*-coordinate System
+% -------------------------------------------------------------------------------------------------------------
+\subsection{Curvilinear \textit{z*}--coordinate System}
+\label{PE_zco_star}
+
+%>>>>>>>>>>>>>>>>>>>>>>>>>>>>
+\begin{figure}[!b] \begin{center}
+\includegraphics[width=1.0\textwidth]{./TexFiles/Figures/Fig_z_zstar.pdf}
+\caption{ \label{Fig_z_zstar}
+(a) $z$-coordinate in linear free-surface case ;
+(b) $z-$coordinate in non-linear free surface case ;
+(c) re-scaled height coordinate (become popular as the \textit{z*-}coordinate
+\citep{Adcroft_Campin_OM04} ).}
+\end{center} \end{figure}
+%>>>>>>>>>>>>>>>>>>>>>>>>>>>>
+
+
+In that case, the free surface equation is nonlinear, and the variations of volume are fully
+taken into account. These coordinates systems is presented in a report \citep{Levier2007}
+available on the \NEMO web site.
+
+%\gmcomment{
+The \textit{z*} coordinate approach is an unapproximated, non-linear free surface implementation
+which allows one to deal with large amplitude free-surface
+variations relative to the vertical resolution \citep{Adcroft_Campin_OM04}. In
+the \textit{z*} formulation, the variation of the column thickness due to sea-surface
+undulations is not concentrated in the surface level, as in the $z$-coordinate formulation,
+but is equally distributed over the full water column. Thus vertical
+levels naturally follow sea-surface variations, with a linear attenuation with
+depth, as illustrated by figure fig.1c . Note that with a flat bottom, such as in
+fig.1c, the bottom-following $z$ coordinate and \textit{z*} are equivalent.
+The definition and modified oceanic equations for the rescaled vertical coordinate
+ \textit{z*}, including the treatment of fresh-water flux at the surface, are
+detailed in Adcroft and Campin (2004). The major points are summarized
+here. The position ( \textit{z*}) and vertical discretization (\textit{z*}) are expressed as:
+\begin{equation} \label{Eq_z-star}
+H + \textit{z*} = (H + z) / r \quad \text{and} \ \delta \textit{z*} = \delta z / r \quad \text{with} \ r = \frac{H+\eta} {H}
+\end{equation}
+Since the vertical displacement of the free surface is incorporated in the vertical
+coordinate \textit{z*}, the upper and lower boundaries are at fixed \textit{z*} position,
+$\textit{z*} = 0$ and $\textit{z*} = -H$ respectively. Also the divergence of the flow field
+is no longer zero as shown by the continuity equation:
+\begin{equation*}
+\frac{\partial r}{\partial t} = \nabla_{\textit{z*}} \cdot \left( r \; \rm{\bf U}_h \right)
+ \left( r \; w\textit{*} \right) = 0
+\end{equation*}
+%}
+
+
+% from MOM4p1 documentation
+
+To overcome problems with vanishing surface and/or bottom cells, we consider the
+zstar coordinate
+\begin{equation} \label{PE_}
+ z^\star = H \left( \frac{z-\eta}{H+\eta} \right)
+\end{equation}
+
+This coordinate is closely related to the "eta" coordinate used in many atmospheric
+models (see Black (1994) for a review of eta coordinate atmospheric models). It
+was originally used in ocean models by Stacey et al. (1995) for studies of tides
+next to shelves, and it has been recently promoted by Adcroft and Campin (2004)
+for global climate modelling.
+
+The surfaces of constant $z^\star$ are quasi-horizontal. Indeed, the $z^\star$ coordinate reduces to $z$ when $\eta$ is zero. In general, when noting the large differences between
+undulations of the bottom topography versus undulations in the surface height, it
+is clear that surfaces constant $z^\star$ are very similar to the depth surfaces. These properties greatly reduce difficulties of computing the horizontal pressure gradient relative to terrain following sigma models discussed in \S\ref{PE_sco}.
+Additionally, since $z^\star$ when $\eta = 0$, no flow is spontaneously generated in an
+unforced ocean starting from rest, regardless the bottom topography. This behaviour is in contrast to the case with "s"-models, where pressure gradient errors in
+the presence of nontrivial topographic variations can generate nontrivial spontaneous flow from a resting state, depending on the sophistication of the pressure
+gradient solver. The quasi-horizontal nature of the coordinate surfaces also facilitates the implementation of neutral physics parameterizations in $z^\star$ models using
+the same techniques as in $z$-models (see Chapters 13-16 of \cite{Griffies_Bk04}) for a
+discussion of neutral physics in $z$-models, as well as Section \S\ref{LDF_slp}
+in this document for treatment in \NEMO).
+
+The range over which $z^\star$ varies is time independent $-H \leq z^\star \leq 0$. Hence, all
+cells remain nonvanishing, so long as the surface height maintains $\eta > ?H$. This
+is a minor constraint relative to that encountered on the surface height when using
+$s = z$ or $s = z - \eta$.
+
+Because $z^\star$ has a time independent range, all grid cells have static increments
+ds, and the sum of the ver tical increments yields the time independent ocean
+depth %·k ds = H.
+The $z^\star$ coordinate is therefore invisible to undulations of the
+free surface, since it moves along with the free surface. This proper ty means that
+no spurious ver tical transpor t is induced across surfaces of constant $z^\star$ by the
+motion of external gravity waves. Such spurious transpor t can be a problem in
+z-models, especially those with tidal forcing. Quite generally, the time independent
+range for the $z^\star$ coordinate is a very convenient proper ty that allows for a nearly
+arbitrary ver tical resolution even in the presence of large amplitude fluctuations of
+the surface height, again so long as $\eta > -H$.
+
+%end MOM doc %%%
+
+
+
+\newpage
+% -------------------------------------------------------------------------------------------------------------
+% Terrain following coordinate System
+% -------------------------------------------------------------------------------------------------------------
+\subsection{Curvilinear Terrain-following \textit{s}--coordinate}
+\label{PE_sco}
+
+% -------------------------------------------------------------------------------------------------------------
+% Introduction
+% -------------------------------------------------------------------------------------------------------------
+\subsubsection{Introduction}
+
+Several important aspects of the ocean circulation are influenced by bottom topography.
+Of course, the most important is that bottom topography determines deep ocean sub-basins,
+barriers, sills and channels that strongly constrain the path of water masses, but more subtle
+effects exist. For example, the topographic $\beta$-effect is usually larger than the planetary
+one along continental slopes. Topographic Rossby waves can be excited and can interact
+with the mean current. In the $z-$coordinate system presented in the previous section
+(\S\ref{PE_zco}), $z-$surfaces are geopotential surfaces. The bottom topography is
+discretised by steps. This often leads to a misrepresentation of a gradually sloping bottom
+and to large localized depth gradients associated with large localized vertical velocities.
+The response to such a velocity field often leads to numerical dispersion effects.
+One solution to strongly reduce this error is to use a partial step representation of bottom
+topography instead of a full step one \cite{Pacanowski_Gnanadesikan_MWR98}.
+Another solution is to introduce a terrain-following coordinate system (hereafter $s-$coordinate)
+
+The $s$-coordinate avoids the discretisation error in the depth field since the layers of
+computation are gradually adjusted with depth to the ocean bottom. Relatively small
+topographic features as well as gentle, large-scale slopes of the sea floor in the deep
+ocean, which would be ignored in typical $z$-model applications with the largest grid
+spacing at greatest depths, can easily be represented (with relatively low vertical resolution).
+A terrain-following model (hereafter $s-$model) also facilitates the modelling of the
+boundary layer flows over a large depth range, which in the framework of the $z$-model
+would require high vertical resolution over the whole depth range. Moreover, with a
+$s$-coordinate it is possible, at least in principle, to have the bottom and the sea surface
+as the only boundaries of the domain (nomore lateral boundary condition to specify).
+Nevertheless, a $s$-coordinate also has its drawbacks. Perfectly adapted to a
+homogeneous ocean, it has strong limitations as soon as stratification is introduced.
+The main two problems come from the truncation error in the horizontal pressure
+gradient and a possibly increased diapycnal diffusion. The horizontal pressure force
+in $s$-coordinate consists of two terms (see Appendix~\ref{Apdx_A}),
+
+\begin{equation} \label{Eq_PE_p_sco}
+\left. {\nabla p} \right|_z =\left. {\nabla p} \right|_s -\frac{\partial
+p}{\partial s}\left. {\nabla z} \right|_s
+\end{equation}
+
+The second term in \eqref{Eq_PE_p_sco} depends on the tilt of the coordinate surface
+and introduces a truncation error that is not present in a $z$-model. In the special case
+of a $\sigma-$coordinate (i.e. a depth-normalised coordinate system $\sigma = z/H$),
+\citet{Haney1991} and \citet{Beckmann1993} have given estimates of the magnitude
+of this truncation error. It depends on topographic slope, stratification, horizontal and
+vertical resolution, the equation of state, and the finite difference scheme. This error
+limits the possible topographic slopes that a model can handle at a given horizontal
+and vertical resolution. This is a severe restriction for large-scale applications using
+realistic bottom topography. The large-scale slopes require high horizontal resolution,
+and the computational cost becomes prohibitive. This problem can be at least partially
+overcome by mixing $s$-coordinate and step-like representation of bottom topography \citep{Gerdes1993a,Gerdes1993b,Madec_al_JPO96}. However, the definition of the model
+domain vertical coordinate becomes then a non-trivial thing for a realistic bottom
+topography: a envelope topography is defined in $s$-coordinate on which a full or
+partial step bottom topography is then applied in order to adjust the model depth to
+the observed one (see \S\ref{DOM_zgr}.
+
+For numerical reasons a minimum of diffusion is required along the coordinate surfaces
+of any finite difference model. It causes spurious diapycnal mixing when coordinate
+surfaces do not coincide with isoneutral surfaces. This is the case for a $z$-model as
+well as for a $s$-model. However, density varies more strongly on $s-$surfaces than
+on horizontal surfaces in regions of large topographic slopes, implying larger diapycnal
+diffusion in a $s$-model than in a $z$-model. Whereas such a diapycnal diffusion in a
+$z$-model tends to weaken horizontal density (pressure) gradients and thus the horizontal
+circulation, it usually reinforces these gradients in a $s$-model, creating spurious circulation.
+For example, imagine an isolated bump of topography in an ocean at rest with a horizontally
+uniform stratification. Spurious diffusion along $s$-surfaces will induce a bump of isoneutral
+surfaces over the topography, and thus will generate there a baroclinic eddy. In contrast,
+the ocean will stay at rest in a $z$-model. As for the truncation error, the problem can be reduced by introducing the terrain-following coordinate below the strongly stratified portion of the water column
+($i.e.$ the main thermocline) \citep{Madec_al_JPO96}. An alternate solution consists of rotating
+the lateral diffusive tensor to geopotential or to isoneutral surfaces (see \S\ref{PE_ldf}.
+Unfortunately, the slope of isoneutral surfaces relative to the $s$-surfaces can very large,
+strongly exceeding the stability limit of such a operator when it is discretized (see Chapter~\ref{LDF}).
+
+The $s-$coordinates introduced here \citep{Lott_al_OM90,Madec_al_JPO96} differ mainly in two
+aspects from similar models: it allows a representation of bottom topography with mixed
+full or partial step-like/terrain following topography ; It also offers a completely general
+transformation, $s=s(i,j,z)$ for the vertical coordinate.
+
+
+\newpage
+% -------------------------------------------------------------------------------------------------------------
+% Curvilinear z-tilde coordinate System
+% -------------------------------------------------------------------------------------------------------------
+\subsection{Curvilinear $\tilde{z}$--coordinate}
+\label{PE_zco_tilde}
+
+The $\tilde{z}$-coordinate has been developed by \citet{Leclair_Madec_OM10s}.
+It is not available in the current version of \NEMO.
+
+\newpage
+% ================================================================
+% Subgrid Scale Physics
+% ================================================================
+\section{Subgrid Scale Physics}
+\label{PE_zdf_ldf}
+
+The primitive equations describe the behaviour of a geophysical fluid at
+space and time scales larger than a few kilometres in the horizontal, a few
+meters in the vertical and a few minutes. They are usually solved at larger
+scales: the specified grid spacing and time step of the numerical model. The
+effects of smaller scale motions (coming from the advective terms in the
+Navier-Stokes equations) must be represented entirely in terms of
+large-scale patterns to close the equations. These effects appear in the
+equations as the divergence of turbulent fluxes ($i.e.$ fluxes associated with
+the mean correlation of small scale perturbations). Assuming a turbulent
+closure hypothesis is equivalent to choose a formulation for these fluxes.
+It is usually called the subgrid scale physics. It must be emphasized that
+this is the weakest part of the primitive equations, but also one of the
+most important for long-term simulations as small scale processes \textit{in fine}
+balance the surface input of kinetic energy and heat.
+
+The control exerted by gravity on the flow induces a strong anisotropy
+between the lateral and vertical motions. Therefore subgrid-scale physics
+\textbf{D}$^{\vect{U}}$, $D^{S}$ and $D^{T}$ in \eqref{Eq_PE_dyn},
+\eqref{Eq_PE_tra_T} and \eqref{Eq_PE_tra_S} are divided into a lateral part
+\textbf{D}$^{l \vect{U}}$, $D^{lS}$ and $D^{lT}$ and a vertical part
+\textbf{D}$^{vU}$, $D^{vS}$ and $D^{vT}$. The formulation of these terms
+and their underlying physics are briefly discussed in the next two subsections.
+
+% -------------------------------------------------------------------------------------------------------------
+% Vertical Subgrid Scale Physics
+% -------------------------------------------------------------------------------------------------------------
+\subsection{Vertical Subgrid Scale Physics}
+\label{PE_zdf}
+
+The model resolution is always larger than the scale at which the major
+sources of vertical turbulence occur (shear instability, internal wave
+breaking...). Turbulent motions are thus never explicitly solved, even
+partially, but always parameterized. The vertical turbulent fluxes are
+assumed to depend linearly on the gradients of large-scale quantities (for
+example, the turbulent heat flux is given by $\overline{T'w'}=-A^{vT} \partial_z \overline T$,
+where $A^{vT}$ is an eddy coefficient). This formulation is
+analogous to that of molecular diffusion and dissipation. This is quite
+clearly a necessary compromise: considering only the molecular viscosity
+acting on large scale severely underestimates the role of turbulent
+diffusion and dissipation, while an accurate consideration of the details of
+turbulent motions is simply impractical. The resulting vertical momentum and
+tracer diffusive operators are of second order:
+\begin{equation} \label{Eq_PE_zdf}
+ \begin{split}
+{\vect{D}}^{v \vect{U}} &=\frac{\partial }{\partial z}\left( {A^{vm}\frac{\partial {\vect{U}}_h }{\partial z}} \right) \ , \\
+D^{vT} &= \frac{\partial }{\partial z}\left( {A^{vT}\frac{\partial T}{\partial z}} \right) \ ,
+\quad
+D^{vS}=\frac{\partial }{\partial z}\left( {A^{vT}\frac{\partial S}{\partial z}} \right)
+ \end{split}
+\end{equation}
+where $A^{vm}$ and $A^{vT}$ are the vertical eddy viscosity and diffusivity coefficients,
+respectively. At the sea surface and at the bottom, turbulent fluxes of momentum, heat
+and salt must be specified (see Chap.~\ref{SBC} and \ref{ZDF} and \S\ref{TRA_bbl}).
+All the vertical physics is embedded in the specification of the eddy coefficients.
+They can be assumed to be either constant, or function of the local fluid properties
+($e.g.$ Richardson number, Brunt-Vais\"{a}l\"{a} frequency...), or computed from a
+turbulent closure model. The choices available in \NEMO are discussed in \S\ref{ZDF}).
+
+% -------------------------------------------------------------------------------------------------------------
+% Lateral Diffusive and Viscous Operators Formulation
+% -------------------------------------------------------------------------------------------------------------
+\subsection{Formulation of the Lateral Diffusive and Viscous Operators}
+\label{PE_ldf}
+
+Lateral turbulence can be roughly divided into a mesoscale turbulence
+associated with eddies (which can be solved explicitly if the resolution is
+sufficient since their underlying physics are included in the primitive
+equations), and a sub mesoscale turbulence which is never explicitly solved
+even partially, but always parameterized. The formulation of lateral eddy
+fluxes depends on whether the mesoscale is below or above the grid-spacing
+($i.e.$ the model is eddy-resolving or not).
+
+In non-eddy-resolving configurations, the closure is similar to that used
+for the vertical physics. The lateral turbulent fluxes are assumed to depend
+linearly on the lateral gradients of large-scale quantities. The resulting
+lateral diffusive and dissipative operators are of second order.
+Observations show that lateral mixing induced by mesoscale turbulence tends
+to be along isopycnal surfaces (or more precisely neutral surfaces \cite{McDougall1987})
+rather than across them.
+As the slope of neutral surfaces is small in the ocean, a common
+approximation is to assume that the `lateral' direction is the horizontal,
+$i.e.$ the lateral mixing is performed along geopotential surfaces. This leads
+to a geopotential second order operator for lateral subgrid scale physics.
+This assumption can be relaxed: the eddy-induced turbulent fluxes can be
+better approached by assuming that they depend linearly on the gradients of
+large-scale quantities computed along neutral surfaces. In such a case,
+the diffusive operator is an isoneutral second order operator and it has
+components in the three space directions. However, both horizontal and
+isoneutral operators have no effect on mean ($i.e.$ large scale) potential
+energy whereas potential energy is a main source of turbulence (through
+baroclinic instabilities). \citet{Gent1990} have proposed a
+parameterisation of mesoscale eddy-induced turbulence which associates an
+eddy-induced velocity to the isoneutral diffusion. Its mean effect is to
+reduce the mean potential energy of the ocean. This leads to a formulation
+of lateral subgrid-scale physics made up of an isoneutral second order
+operator and an eddy induced advective part. In all these lateral diffusive
+formulations, the specification of the lateral eddy coefficients remains the
+problematic point as there is no really satisfactory formulation of these
+coefficients as a function of large-scale features.
+
+In eddy-resolving configurations, a second order operator can be used, but
+usually the more scale selective biharmonic operator is preferred as the
+grid-spacing is usually not small enough compared to the scale of the
+eddies. The role devoted to the subgrid-scale physics is to dissipate the
+energy that cascades toward the grid scale and thus to ensure the stability of
+the model while not interfering with the resolved mesoscale activity. Another approach
+is becoming more and more popular: instead of specifying explicitly a sub-grid scale
+term in the momentum and tracer time evolution equations, one uses a advective
+scheme which is diffusive enough to maintain the model stability. It must be emphasised
+that then, all the sub-grid scale physics is included in the formulation of the
+advection scheme.
+
+All these parameterisations of subgrid scale physics have advantages and
+drawbacks. There are not all available in \NEMO. In the $z$-coordinate
+formulation, five options are offered for active tracers (temperature and
+salinity): second order geopotential operator, second order isoneutral
+operator, \citet{Gent1990} parameterisation, fourth order
+geopotential operator, and various slightly diffusive advection schemes.
+The same options are available for momentum, except
+\citet{Gent1990} parameterisation which only involves tracers. In the
+$s$-coordinate formulation, additional options are offered for tracers: second
+order operator acting along $s-$surfaces, and for momentum: fourth order
+operator acting along $s-$surfaces (see \S\ref{LDF}).
+
+\subsubsection{Lateral second order tracer diffusive operator}
+
+The lateral second order tracer diffusive operator is defined by (see Appendix~\ref{Apdx_B}):
+\begin{equation} \label{Eq_PE_iso_tensor}
+D^{lT}=\nabla {\rm {\bf .}}\left( {A^{lT}\;\Re \;\nabla T} \right) \qquad
+\mbox{with}\quad \;\;\Re =\left( {{\begin{array}{*{20}c}
+ 1 \hfill & 0 \hfill & {-r_1 } \hfill \\
+ 0 \hfill & 1 \hfill & {-r_2 } \hfill \\
+ {-r_1 } \hfill & {-r_2 } \hfill & {r_1 ^2+r_2 ^2} \hfill \\
+\end{array} }} \right)
+\end{equation}
+where $r_1 \;\mbox{and}\;r_2 $ are the slopes between the surface along
+which the diffusive operator acts and the model level ($e. g.$ $z$- or
+$s$-surfaces). Note that the formulation \eqref{Eq_PE_iso_tensor} is exact for the
+rotation between geopotential and $s$-surfaces, while it is only an approximation
+for the rotation between isoneutral and $z$- or $s$-surfaces. Indeed, in the latter
+case, two assumptions are made to simplify \eqref{Eq_PE_iso_tensor} \citep{Cox1987}.
+First, the horizontal contribution of the dianeutral mixing is neglected since the ratio
+between iso and dia-neutral diffusive coefficients is known to be several orders of
+magnitude smaller than unity. Second, the two isoneutral directions of diffusion are
+assumed to be independent since the slopes are generally less than $10^{-2}$ in the
+ocean (see Appendix~\ref{Apdx_B}).
+
+For \textit{geopotential} diffusion, $r_1$ and $r_2 $ are the slopes between the
+geopotential and computational surfaces: in $z$-coordinates they are zero
+($r_1 = r_2 = 0$) while in $s$-coordinate (including $\textit{z*}$ case) they are
+equal to $\sigma _1$ and $\sigma _2$, respectively (see \eqref{Eq_PE_sco_slope} ).
+
+For \textit{isoneutral} diffusion $r_1$ and $r_2$ are the slopes between the isoneutral
+and computational surfaces. Therefore, they are different quantities,
+but have similar expressions in $z$- and $s$-coordinates. In $z$-coordinates:
+\begin{equation} \label{Eq_PE_iso_slopes}
+r_1 =\frac{e_3 }{e_1 } \left( {\frac{\partial \rho }{\partial i}} \right)
+ \left( {\frac{\partial \rho }{\partial k}} \right)^{-1} \ , \quad
+r_1 =\frac{e_3 }{e_1 } \left( {\frac{\partial \rho }{\partial i}} \right)
+ \left( {\frac{\partial \rho }{\partial k}} \right)^{-1},
+\end{equation}
+while in $s$-coordinates $\partial/\partial k$ is replaced by
+$\partial/\partial s$.
+
+\subsubsection{Eddy induced velocity}
+ When the \textit{eddy induced velocity} parametrisation (eiv) \citep{Gent1990} is used,
+an additional tracer advection is introduced in combination with the isoneutral diffusion of tracers:
+\begin{equation} \label{Eq_PE_iso+eiv}
+D^{lT}=\nabla \cdot \left( {A^{lT}\;\Re \;\nabla T} \right)
+ +\nabla \cdot \left( {{\vect{U}}^\ast \,T} \right)
+\end{equation}
+where ${\vect{U}}^\ast =\left( {u^\ast ,v^\ast ,w^\ast } \right)$ is a non-divergent,
+eddy-induced transport velocity. This velocity field is defined by:
+\begin{equation} \label{Eq_PE_eiv}
+ \begin{split}
+ u^\ast &= +\frac{1}{e_3 }\frac{\partial }{\partial k}\left[ {A^{eiv}\;\tilde{r}_1 } \right] \\
+ v^\ast &= +\frac{1}{e_3 }\frac{\partial }{\partial k}\left[ {A^{eiv}\;\tilde{r}_2 } \right] \\
+ w^\ast &= -\frac{1}{e_1 e_2 }\left[
+ \frac{\partial }{\partial i}\left( {A^{eiv}\;e_2\,\tilde{r}_1 } \right)
+ +\frac{\partial }{\partial j}\left( {A^{eiv}\;e_1\,\tilde{r}_2 } \right) \right]
+ \end{split}
+\end{equation}
+where $A^{eiv}$ is the eddy induced velocity coefficient (or equivalently the isoneutral
+thickness diffusivity coefficient), and $\tilde{r}_1$ and $\tilde{r}_2$ are the slopes
+between isoneutral and \emph{geopotential} surfaces. Their values are
+thus independent of the vertical coordinate, but their expression depends on the coordinate:
+\begin{align} \label{Eq_PE_slopes_eiv}
+\tilde{r}_n = \begin{cases}
+ r_n & \text{in $z$-coordinate} \\
+ r_n + \sigma_n & \text{in \textit{z*} and $s$-coordinates}
+ \end{cases}
+\quad \text{where } n=1,2
+\end{align}
+
+The normal component of the eddy induced velocity is zero at all the boundaries.
+This can be achieved in a model by tapering either the eddy coefficient or the slopes
+to zero in the vicinity of the boundaries. The latter strategy is used in \NEMO (cf. Chap.~\ref{LDF}).
+
+\subsubsection{Lateral fourth order tracer diffusive operator}
+
+The lateral fourth order tracer diffusive operator is defined by:
+\begin{equation} \label{Eq_PE_bilapT}
+D^{lT}=\Delta \left( {A^{lT}\;\Delta T} \right)
+\qquad \text{where} \ D^{lT}=\Delta \left( {A^{lT}\;\Delta T} \right)
+ \end{equation}
+
+It is the second order operator given by \eqref{Eq_PE_iso_tensor} applied twice with
+the eddy diffusion coefficient correctly placed.
+
+
+\subsubsection{Lateral second order momentum diffusive operator}
+
+The second order momentum diffusive operator along $z$- or $s$-surfaces is found by
+applying \eqref{Eq_PE_lap_vector} to the horizontal velocity vector (see Appendix~\ref{Apdx_B}):
+\begin{equation} \label{Eq_PE_lapU}
+\begin{split}
+{\rm {\bf D}}^{l{\rm {\bf U}}}
+&= \quad \ \nabla _h \left( {A^{lm}\chi } \right)
+ \ - \ \nabla _h \times \left( {A^{lm}\,\zeta \;{\rm {\bf k}}} \right) \\
+&= \left( \begin{aligned}
+ \frac{1}{e_1 } \frac{\partial \left( A^{lm} \chi \right)}{\partial i}
+ &-\frac{1}{e_2 e_3}\frac{\partial \left( {A^{lm} \;e_3 \zeta} \right)}{\partial j} \\
+ \frac{1}{e_2 }\frac{\partial \left( {A^{lm} \chi } \right)}{\partial j}
+ &+\frac{1}{e_1 e_3}\frac{\partial \left( {A^{lm} \;e_3 \zeta} \right)}{\partial i}
+ \end{aligned} \right)
+\end{split}
+\end{equation}
+
+Such a formulation ensures a complete separation between the vorticity and
+horizontal divergence fields (see Appendix~\ref{Apdx_C}). Unfortunately, it is not
+available for geopotential diffusion in $s-$coordinates and for isoneutral
+diffusion in both $z$- and $s$-coordinates ($i.e.$ when a rotation is required).
+In these two cases, the $u$ and $v-$fields are considered as independent scalar
+fields, so that the diffusive operator is given by:
+\begin{equation} \label{Eq_PE_lapU_iso}
+\begin{split}
+ D_u^{l{\rm {\bf U}}} &= \nabla .\left( {\Re \;\nabla u} \right) \\
+ D_v^{l{\rm {\bf U}}} &= \nabla .\left( {\Re \;\nabla v} \right)
+ \end{split}
+ \end{equation}
+where $\Re$ is given by \eqref{Eq_PE_iso_tensor}. It is the same expression as
+those used for diffusive operator on tracers. It must be emphasised that such a
+formulation is only exact in a Cartesian coordinate system, $i.e.$ on a $f-$ or
+$\beta-$plane, not on the sphere. It is also a very good approximation in vicinity
+of the Equator in a geographical coordinate system \citep{Lengaigne_al_JGR03}.
+
+\subsubsection{lateral fourth order momentum diffusive operator}
+
+As for tracers, the fourth order momentum diffusive operator along $z$ or $s$-surfaces
+is a re-entering second order operator \eqref{Eq_PE_lapU} or \eqref{Eq_PE_lapU}
+with the eddy viscosity coefficient correctly placed:
+
+geopotential diffusion in $z$-coordinate:
+\begin{equation} \label{Eq_PE_bilapU}
+\begin{split}
+{\rm {\bf D}}^{l{\rm {\bf U}}} &=\nabla _h \left\{ {\;\nabla _h {\rm {\bf
+.}}\left[ {A^{lm}\,\nabla _h \left( \chi \right)} \right]\;}
+\right\}\; \\
+&+\nabla _h \times \left\{ {\;{\rm {\bf k}}\cdot \nabla \times
+\left[ {A^{lm}\,\nabla _h \times \left( {\zeta \;{\rm {\bf k}}} \right)}
+\right]\;} \right\}
+\end{split}
+\end{equation}
+
+\gmcomment{ change the position of the coefficient, both here and in the code}
+
+geopotential diffusion in $s$-coordinate:
+\begin{equation} \label{Eq_bilapU_iso}
+ \left\{ \begin{aligned}
+ D_u^{l{\rm {\bf U}}} =\Delta \left( {A^{lm}\;\Delta u} \right) \\
+ D_v^{l{\rm {\bf U}}} =\Delta \left( {A^{lm}\;\Delta v} \right)
+ \end{aligned} \right.
+ \quad \text{where} \quad
+ \Delta \left( \bullet \right) = \nabla \cdot \left( \Re \nabla(\bullet) \right)
+\end{equation}
+
Index: /branches/UKMO/dev_r5518_GO6_under_ice_relax_dr_hook/DOC/TexFiles/Chapters/Chap_Model_Basics_zstar.tex
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+% ================================================================
+% Chapter 1 � Model Basics
+% ================================================================
+% ================================================================
+% Curvilinear z*- s*-coordinate System
+% ================================================================
+\chapter{ essai z* s*}
+\section{Curvilinear \textit{z*}- or \textit{s*} coordinate System}
+
+% -------------------------------------------------------------------------------------------------------------
+% ????
+% -------------------------------------------------------------------------------------------------------------
+
+\colorbox{yellow}{ to be updated }
+
+In that case, the free surface equation is nonlinear, and the variations of
+volume are fully taken into account. These coordinates systems is presented in
+a report \citep{Levier2007} available on the \NEMO web site.
+
+\colorbox{yellow}{ end of to be updated}
+\newline
+
+% from MOM4p1 documentation
+
+To overcome problems with vanishing surface and/or bottom cells, we consider the
+zstar coordinate
+\begin{equation} \label{PE_}
+ z^\star = H \left( \frac{z-\eta}{H+\eta} \right)
+\end{equation}
+
+This coordinate is closely related to the "eta" coordinate used in many atmospheric
+models (see Black (1994) for a review of eta coordinate atmospheric models). It
+was originally used in ocean models by Stacey et al. (1995) for studies of tides
+next to shelves, and it has been recently promoted by Adcroft and Campin (2004)
+for global climate modelling.
+
+The surfaces of constant $z^\star$ are quasi-horizontal. Indeed, the $z^\star$ coordinate reduces to $z$ when $\eta$ is zero. In general, when noting the large differences between
+undulations of the bottom topography versus undulations in the surface height, it
+is clear that surfaces constant $z^\star$ are very similar to the depth surfaces. These properties greatly reduce difficulties of computing the horizontal pressure gradient relative to terrain following sigma models discussed in \S\ref{PE_sco}.
+Additionally, since $z^\star$ when $\eta = 0$, no flow is spontaneously generated in an
+unforced ocean starting from rest, regardless the bottom topography. This behaviour is in contrast to the case with "s"-models, where pressure gradient errors in
+the presence of nontrivial topographic variations can generate nontrivial spontaneous flow from a resting state, depending on the sophistication of the pressure
+gradient solver. The quasi-horizontal nature of the coordinate surfaces also facilitates the implementation of neutral physics parameterizations in $z^\star$ models using
+the same techniques as in $z$-models (see Chapters 13-16 of Griffies (2004) for a
+discussion of neutral physics in $z$-models, as well as Section \S\ref{LDF_slp}
+in this document for treatment in \NEMO).
+
+The range over which $z^\star$ varies is time independent $-H \leq z^\star \leq 0$. Hence, all
+cells remain nonvanishing, so long as the surface height maintains $\eta > ?H$. This
+is a minor constraint relative to that encountered on the surface height when using
+$s = z$ or $s = z - \eta$.
+
+Because $z^\star$ has a time independent range, all grid cells have static increments
+ds, and the sum of the ver tical increments yields the time independent ocean
+depth %�k ds = H.
+The $z^\star$ coordinate is therefore invisible to undulations of the
+free surface, since it moves along with the free surface. This proper ty means that
+no spurious ver tical transpor t is induced across surfaces of constant $z^\star$ by the
+motion of external gravity waves. Such spurious transpor t can be a problem in
+z-models, especially those with tidal forcing. Quite generally, the time independent
+range for the $z^\star$ coordinate is a very convenient proper ty that allows for a nearly
+arbitrary ver tical resolution even in the presence of large amplitude fluctuations of
+the surface height, again so long as $\eta > -H$.
+
+
+
+%%%
+% essai update time splitting...
+%%%
+
+
+% ================================================================
+% Surface Pressure Gradient and Sea Surface Height
+% ================================================================
+\section{Surface pressure gradient and Sea Surface Heigth (\mdl{dynspg})}
+\label{DYN_hpg_spg}
+%-----------------------------------------nam_dynspg----------------------------------------------------
+\namdisplay{nam_dynspg}
+%------------------------------------------------------------------------------------------------------------
+Options are defined through the \ngn{nam\_dynspg} namelist variables.
+The surface pressure gradient term is related to the representation of the free surface (\S\ref{PE_hor_pg}). The main distinction is between the fixed volume case (linear free surface or rigid lid) and the variable volume case (nonlinear free surface, \key{vvl} is active). In the linear free surface case (\S\ref{PE_free_surface}) and rigid lid (\S\ref{PE_rigid_lid}), the vertical scale factors $e_{3}$ are fixed in time, while in the nonlinear case (\S\ref{PE_free_surface}) they are time-dependent. With both linear and nonlinear free surface, external gravity waves are allowed in the equations, which imposes a very small time step when an explicit time stepping is used. Two methods are proposed to allow a longer time step for the three-dimensional equations: the filtered free surface, which is a modification of the continuous equations (see \eqref{Eq_PE_flt}), and the split-explicit free surface described below. The extra term introduced in the filtered method is calculated implicitly, so that the update of the next velocities is done in module \mdl{dynspg\_flt} and not in \mdl{dynnxt}.
+
+%-------------------------------------------------------------
+% Explicit
+%-------------------------------------------------------------
+\subsubsection{Explicit (\key{dynspg\_exp})}
+\label{DYN_spg_exp}
+
+In the explicit free surface formulation, the model time step is chosen small enough to describe the external gravity waves (typically a few ten seconds). The sea surface height is given by :
+\begin{equation} \label{Eq_dynspg_ssh}
+\frac{\partial \eta }{\partial t}\equiv \frac{\text{EMP}}{\rho _w }+\frac{1}{e_{1T}
+e_{2T} }\sum\limits_k {\left( {\delta _i \left[ {e_{2u} e_{3u} u}
+\right]+\delta _j \left[ {e_{1v} e_{3v} v} \right]} \right)}
+\end{equation}
+
+where EMP is the surface freshwater budget (evaporation minus precipitation, and minus river runoffs (if the later are introduced as a surface freshwater flux, see \S\ref{SBC}) expressed in $Kg.m^{-2}.s^{-1}$, and $\rho _w =1,000\,Kg.m^{-3}$ is the volumic mass of pure water. The sea-surface height is evaluated using a leapfrog scheme in combination with an Asselin time filter, i.e. the velocity appearing in (\ref{Eq_dynspg_ssh}) is centred in time (\textit{now} velocity).
+
+The surface pressure gradient, also evaluated using a leap-frog scheme, is then simply given by :
+\begin{equation} \label{Eq_dynspg_exp}
+\left\{ \begin{aligned}
+ - \frac{1} {e_{1u}} \; \delta _{i+1/2} \left[ \,\eta\, \right] \\
+ \\
+ - \frac{1} {e_{2v}} \; \delta _{j+1/2} \left[ \,\eta\, \right]
+\end{aligned} \right.
+\end{equation}
+
+Consistent with the linearization, a $\left. \rho \right|_{k=1} / \rho _o$ factor is omitted in (\ref{Eq_dynspg_exp}).
+
+%-------------------------------------------------------------
+% Split-explicit time-stepping
+%-------------------------------------------------------------
+\subsubsection{Split-explicit time-stepping (\key{dynspg\_ts})}
+\label{DYN_spg_ts}
+%--------------------------------------------namdom----------------------------------------------------
+\namdisplay{namdom}
+%--------------------------------------------------------------------------------------------------------------
+The split-explicit free surface formulation used in OPA follows the one proposed by \citet{Griffies2004}. The general idea is to solve the free surface equation with a small time step, while the three dimensional prognostic variables are solved with a longer time step that is a multiple of \np{rdtbt}
+in the \ngn{namdom} namelist.
+(Figure III.3).
+
+%> > > > > > > > > > > > > > > > > > > > > > > > > > > >
+\begin{figure}[!t] \begin{center}
+\includegraphics[width=0.90\textwidth]{./Figures/Fig_DYN_dynspg_ts.pdf}
+\caption{ \label{Fig_DYN_dynspg_ts}
+Schematic of the split-explicit time stepping scheme for the barotropic and baroclinic modes,
+after \citet{Griffies2004}. Time increases to the right. Baroclinic time steps are denoted by
+$t-\Delta t$, $t, t+\Delta t$, and $t+2\Delta t$. The curved line represents a leap-frog time step,
+and the smaller barotropic time steps $N \Delta t=2\Delta t$ are denoted by the zig-zag line.
+The vertically integrated forcing \textbf{M}(t) computed at baroclinic time step t represents
+the interaction between the barotropic and baroclinic motions. While keeping the total depth,
+tracer, and freshwater forcing fields fixed, a leap-frog integration carries the surface height
+and vertically integrated velocity from t to $t+2 \Delta t$ using N barotropic time steps of length
+$\Delta t$. Time averaging the barotropic fields over the N+1 time steps (endpoints included)
+centers the vertically integrated velocity at the baroclinic timestep $t+\Delta t$.
+A baroclinic leap-frog time step carries the surface height to $t+\Delta t$ using the convergence
+of the time averaged vertically integrated velocity taken from baroclinic time step t. }
+\end{center}
+\end{figure}
+%> > > > > > > > > > > > > > > > > > > > > > > > > > > >
+
+The split-explicit formulation has a damping effect on external gravity waves, which is weaker than the filtered free surface but still significant as shown by \citet{Levier2007} in the case of an analytical barotropic Kelvin wave.
+
+%from griffies book: ..... copy past !
+
+\textbf{title: Time stepping the barotropic system }
+
+Assume knowledge of the full velocity and tracer fields at baroclinic time $\tau$. Hence,
+we can update the surface height and vertically integrated velocity with a leap-frog
+scheme using the small barotropic time step $\Delta t$. We have
+
+\begin{equation} \label{DYN_spg_ts_eta}
+\eta^{(b)}(\tau,t_{n+1}) - \eta^{(b)}(\tau,t_{n+1}) (\tau,t_{n-1})
+ = 2 \Delta t \left[-\nabla \cdot \textbf{U}^{(b)}(\tau,t_n) + \text{EMP}_w(\tau) \right]
+\end{equation}
+\begin{multline} \label{DYN_spg_ts_u}
+\textbf{U}^{(b)}(\tau,t_{n+1}) - \textbf{U}^{(b)}(\tau,t_{n-1}) \\
+ = 2\Delta t \left[ - f \textbf{k} \times \textbf{U}^{(b)}(\tau,t_{n})
+ - H(\tau) \nabla p_s^{(b)}(\tau,t_{n}) +\textbf{M}(\tau) \right]
+\end{multline}
+\
+
+In these equations, araised (b) denotes values of surface height and vertically integrated velocity updated with the barotropic time steps. The $\tau$ time label on $\eta^{(b)}$
+and $U^{(b)}$ denotes the baroclinic time at which the vertically integrated forcing $\textbf{M}(\tau)$ (note that this forcing includes the surface freshwater forcing), the tracer fields, the freshwater flux $\text{EMP}_w(\tau)$, and total depth of the ocean $H(\tau)$ are held for the duration of the barotropic time stepping over a single cycle. This is also the time
+that sets the barotropic time steps via
+\begin{equation} \label{DYN_spg_ts_t}
+t_n=\tau+n\Delta t
+\end{equation}
+with $n$ an integer. The density scaled surface pressure is evaluated via
+\begin{equation} \label{DYN_spg_ts_ps}
+p_s^{(b)}(\tau,t_{n}) = \begin{cases}
+ g \;\eta_s^{(b)}(\tau,t_{n}) \;\rho(\tau)_{k=1}) / \rho_o & \text{non-linear case} \\
+ g \;\eta_s^{(b)}(\tau,t_{n}) & \text{linear case}
+ \end{cases}
+\end{equation}
+To get started, we assume the following initial conditions
+\begin{equation} \label{DYN_spg_ts_eta}
+\begin{split}
+\eta^{(b)}(\tau,t_{n=0}) &= \overline{\eta^{(b)}(\tau)}
+\\
+\eta^{(b)}(\tau,t_{n=1}) &= \eta^{(b)}(\tau,t_{n=0}) + \Delta t \ \text{RHS}_{n=0}
+\end{split}
+\end{equation}
+with
+\begin{equation} \label{DYN_spg_ts_etaF}
+ \overline{\eta^{(b)}(\tau)} = \frac{1}{N+1} \sum\limits_{n=0}^N \eta^{(b)}(\tau-\Delta t,t_{n})
+\end{equation}
+the time averaged surface height taken from the previous barotropic cycle. Likewise,
+\begin{equation} \label{DYN_spg_ts_u}
+\textbf{U}^{(b)}(\tau,t_{n=0}) = \overline{\textbf{U}^{(b)}(\tau)} \\
+\\
+\textbf{U}(\tau,t_{n=1}) = \textbf{U}^{(b)}(\tau,t_{n=0}) + \Delta t \ \text{RHS}_{n=0}
+\end{equation}
+with
+\begin{equation} \label{DYN_spg_ts_u}
+ \overline{\textbf{U}^{(b)}(\tau)}
+ = \frac{1}{N+1} \sum\limits_{n=0}^N\textbf{U}^{(b)}(\tau-\Delta t,t_{n})
+\end{equation}
+the time averaged vertically integrated transport. Notably, there is no Robert-Asselin time filter used in the barotropic portion of the integration.
+
+Upon reaching $t_{n=N} = \tau + 2\Delta \tau$ , the vertically integrated velocity is time averaged to produce the updated vertically integrated velocity at baroclinic time $\tau + \Delta \tau$
+\begin{equation} \label{DYN_spg_ts_u}
+\textbf{U}(\tau+\Delta t) = \overline{\textbf{U}^{(b)}(\tau+\Delta t)}
+ = \frac{1}{N+1} \sum\limits_{n=0}^N\textbf{U}^{(b)}(\tau,t_{n})
+\end{equation}
+The surface height on the new baroclinic time step is then determined via a baroclinic leap-frog using the following form
+
+\begin{equation} \label{DYN_spg_ts_ssh}
+\eta(\tau+\Delta) - \eta^{F}(\tau-\Delta) = 2\Delta t \ \left[ - \nabla \cdot \textbf{U}(\tau) + \text{EMP}_w \right]
+\end{equation}
+
+ The use of this "big-leap-frog" scheme for the surface height ensures compatibility between the mass/volume budgets and the tracer budgets. More discussion of this point is provided in Chapter 10 (see in particular Section 10.2).
+
+In general, some form of time filter is needed to maintain integrity of the surface
+height field due to the leap-frog splitting mode in equation \ref{DYN_spg_ts_ssh}. We
+have tried various forms of such filtering, with the following method discussed in
+Griffies et al. (2001) chosen due to its stability and reasonably good maintenance of
+tracer conservation properties (see Section ??)
+
+\begin{equation} \label{DYN_spg_ts_sshf}
+\eta^{F}(\tau-\Delta) = \overline{\eta^{(b)}(\tau)}
+\end{equation}
+Another approach tried was
+
+\begin{equation} \label{DYN_spg_ts_sshf2}
+\eta^{F}(\tau-\Delta) = \eta(\tau)
+ + (\alpha/2) \left[\overline{\eta^{(b)}}(\tau+\Delta t)
+ + \overline{\eta^{(b)}}(\tau-\Delta t) -2 \;\eta(\tau) \right]
+\end{equation}
+
+which is useful since it isolates all the time filtering aspects into the term multiplied
+by $\alpha$. This isolation allows for an easy check that tracer conservation is exact when
+eliminating tracer and surface height time filtering (see Section ?? for more complete discussion). However, in the general case with a non-zero $\alpha$, the filter \ref{DYN_spg_ts_sshf} was found to be more conservative, and so is recommended.
+
+
+
+
+
+%-------------------------------------------------------------
+% Filtered formulation
+%-------------------------------------------------------------
+\subsubsection{Filtered formulation (\key{dynspg\_flt})}
+\label{DYN_spg_flt}
+
+The filtered formulation follows the \citet{Roullet2000} implementation. The extra term introduced in the equations (see {\S}I.2.2) is solved implicitly. The elliptic solvers available in the code are
+documented in \S\ref{MISC}. The amplitude of the extra term is given by the namelist variable \np{rnu}. The default value is 1, as recommended by \citet{Roullet2000}
+
+\colorbox{red}{\np{rnu}=1 to be suppressed from namelist !}
+
+%-------------------------------------------------------------
+% Non-linear free surface formulation
+%-------------------------------------------------------------
+\subsection{Non-linear free surface formulation (\key{vvl})}
+\label{DYN_spg_vvl}
+
+In the non-linear free surface formulation, the variations of volume are fully taken into account. This option is presented in a report \citep{Levier2007} available on the NEMO web site. The three time-stepping methods (explicit, split-explicit and filtered) are the same as in \S\ref{DYN_spg_linear} except that the ocean depth is now time-dependent. In particular, this means that in filtered case, the matrix to be inverted has to be recomputed at each time-step.
+
+
Index: /branches/UKMO/dev_r5518_GO6_under_ice_relax_dr_hook/DOC/TexFiles/Chapters/Chap_OBS.tex
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+% ================================================================
+% Chapter observation operator (OBS)
+% ================================================================
+\chapter{Observation and model comparison (OBS)}
+\label{OBS}
+
+Authors: D. Lea, M. Martin, K. Mogensen, A. Vidard, A. Weaver, A. Ryan, ... % do we keep that ?
+
+\minitoc
+
+
+\newpage
+$\ $\newline % force a new line
+
+The observation and model comparison code (OBS) reads in observation files (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 timestep. The resulting data are saved in a ``feedback'' file (or
+files). The code was originally developed for use with the NEMOVAR data assimilation code, but
+can be used for validation or verification of model or any other data assimilation system.
+
+The OBS code is called from \mdl{nemogcm.F90} for model initialisation and to calculate the model
+equivalent values for observations on the 0th timestep. The code is then called again after
+each timestep from \mdl{step.F90}. To build with the OBS code active \key{diaobs} must be
+set.
+
+For all data types a 2D horizontal interpolator is needed to interpolate the model fields to
+the observation location. For {\em in situ} profiles, a 1D vertical interpolator is needed in
+addition to provide model fields at the observation depths. Currently this only works in
+z-level model configurations, but is being developed to work with a generalised vertical
+coordinate system. Temperature data from moored buoys (TAO, TRITON, PIRATA) in the
+ENACT/ENSEMBLES data-base are available as daily averaged quantities. For this type of
+observation the observation operator will compare such observations to the model temperature
+fields averaged over one day. The relevant observation type may be specified in the namelist
+using \np{endailyavtypes}. Otherwise the model value from the nearest timestep to the
+observation time is used.
+
+The code is controlled by the namelist \textit{nam\_obs}. See the following sections for more
+details on setting up the namelist.
+
+Section~\ref{OBS_example} introduces a test example of the observation operator code including
+where to obtain data and how to setup the namelist. Section~\ref{OBS_details} introduces some
+more technical details of the different observation types used and also shows a more complete
+namelist. Section~\ref{OBS_theory} introduces some of the theoretical aspects of the observation
+operator including interpolation methods and running on multiple processors.
+Section~\ref{OBS_ooo} describes the offline observation operator code.
+Section~\ref{OBS_obsutils} introduces some utilities to help working with the files
+produced by the OBS code.
+
+% ================================================================
+% Example
+% ================================================================
+\section{Running the observation operator code example}
+\label{OBS_example}
+
+This section describes an example of running the observation operator code using
+profile data which can be freely downloaded. It shows how to adapt an
+existing run and build of NEMO to run the observation operator.
+
+\begin{enumerate}
+\item Compile NEMO with \key{diaobs} set.
+
+\item Download some ENSEMBLES EN3 data from
+\href{http://www.hadobs.org}{http://www.hadobs.org}. Choose observations which are
+valid for the period of your test run because the observation operator compares
+the model and observations for a matching date and time.
+
+\item Add the following to the NEMO namelist to run the observation
+operator on this data. Set the \np{enactfiles} namelist variable to the
+observation file name:
+\end{enumerate}
+
+%------------------------------------------namobs_example-----------------------------------------------------
+\namdisplay{namobs_example}
+%-------------------------------------------------------------------------------------------------------------
+
+Options are defined through the \ngn{namobs} namelist variables.
+The options \np{ln\_t3d} and \np{ln\_s3d} switch on the temperature and salinity
+profile observation operator code. The \np{ln\_ena} switch turns on the reading
+of ENACT/ENSEMBLES type profile data. The filename or array of filenames are
+specified using the \np{enactfiles} variable. The model grid points for a
+particular observation latitude and longitude are found using the grid
+searching part of the code. This can be expensive, particularly for large
+numbers of observations, setting \np{ln\_grid\_search\_lookup} allows the use of
+a lookup table which is saved into an ``xypos`` file (or files). This will need
+to be generated the first time if it does not exist in the run directory.
+However, once produced it will significantly speed up future grid searches.
+Setting \np{ln\_grid\_global} means that the code distributes the observations
+evenly between processors. Alternatively each processor will work with
+observations located within the model subdomain (see section~\ref{OBS_parallel}).
+
+A number of utilities are now provided to plot the feedback files, convert and
+recombine the files. These are explained in more detail in section~\ref{OBS_obsutils}.
+
+\section{Technical details}
+\label{OBS_details}
+
+Here we show a more complete example namelist \ngn{namobs} and also show the NetCDF headers
+of the observation
+files that may be used with the observation operator
+
+%------------------------------------------namobs--------------------------------------------------------
+\namdisplay{namobs}
+%-------------------------------------------------------------------------------------------------------------
+
+This name list uses the "feedback" type observation file input format for
+profile, sea level anomaly and sea surface temperature data. All the
+observation files must be in NetCDF format. Some example headers (produced using
+\mbox{\textit{ncdump~-h}}) for profile
+data, sea level anomaly and sea surface temperature are in the following
+subsections.
+
+\subsection{Profile feedback type observation file header}
+
+\begin{alltt}
+\tiny
+\begin{verbatim}
+netcdf profiles_01 {
+dimensions:
+ N_OBS = 603 ;
+ N_LEVELS = 150 ;
+ N_VARS = 2 ;
+ N_QCF = 2 ;
+ N_ENTRIES = 1 ;
+ N_EXTRA = 1 ;
+ STRINGNAM = 8 ;
+ STRINGGRID = 1 ;
+ STRINGWMO = 8 ;
+ STRINGTYP = 4 ;
+ STRINGJULD = 14 ;
+variables:
+ char VARIABLES(N_VARS, STRINGNAM) ;
+ VARIABLES:long_name = "List of variables in feedback files" ;
+ char ENTRIES(N_ENTRIES, STRINGNAM) ;
+ ENTRIES:long_name = "List of additional entries for each variable in feedback files" ;
+ char EXTRA(N_EXTRA, STRINGNAM) ;
+ EXTRA:long_name = "List of extra variables" ;
+ char STATION_IDENTIFIER(N_OBS, STRINGWMO) ;
+ STATION_IDENTIFIER:long_name = "Station identifier" ;
+ char STATION_TYPE(N_OBS, STRINGTYP) ;
+ STATION_TYPE:long_name = "Code instrument type" ;
+ double LONGITUDE(N_OBS) ;
+ LONGITUDE:long_name = "Longitude" ;
+ LONGITUDE:units = "degrees_east" ;
+ LONGITUDE:_Fillvalue = 99999.f ;
+ double LATITUDE(N_OBS) ;
+ LATITUDE:long_name = "Latitude" ;
+ LATITUDE:units = "degrees_north" ;
+ LATITUDE:_Fillvalue = 99999.f ;
+ double DEPTH(N_OBS, N_LEVELS) ;
+ DEPTH:long_name = "Depth" ;
+ DEPTH:units = "metre" ;
+ DEPTH:_Fillvalue = 99999.f ;
+ int DEPTH_QC(N_OBS, N_LEVELS) ;
+ DEPTH_QC:long_name = "Quality on depth" ;
+ DEPTH_QC:Conventions = "q where q =[0,9]" ;
+ DEPTH_QC:_Fillvalue = 0 ;
+ int DEPTH_QC_FLAGS(N_OBS, N_LEVELS, N_QCF) ;
+ DEPTH_QC_FLAGS:long_name = "Quality flags on depth" ;
+ DEPTH_QC_FLAGS:Conventions = "NEMOVAR flag conventions" ;
+ double JULD(N_OBS) ;
+ JULD:long_name = "Julian day" ;
+ JULD:units = "days since JULD_REFERENCE" ;
+ JULD:Conventions = "relative julian days with decimal part (as parts of day)" ;
+ JULD:_Fillvalue = 99999.f ;
+ char JULD_REFERENCE(STRINGJULD) ;
+ JULD_REFERENCE:long_name = "Date of reference for julian days" ;
+ JULD_REFERENCE:Conventions = "YYYYMMDDHHMMSS" ;
+ int OBSERVATION_QC(N_OBS) ;
+ OBSERVATION_QC:long_name = "Quality on observation" ;
+ OBSERVATION_QC:Conventions = "q where q =[0,9]" ;
+ OBSERVATION_QC:_Fillvalue = 0 ;
+ int OBSERVATION_QC_FLAGS(N_OBS, N_QCF) ;
+ OBSERVATION_QC_FLAGS:long_name = "Quality flags on observation" ;
+ OBSERVATION_QC_FLAGS:Conventions = "NEMOVAR flag conventions" ;
+ OBSERVATION_QC_FLAGS:_Fillvalue = 0 ;
+ int POSITION_QC(N_OBS) ;
+ POSITION_QC:long_name = "Quality on position (latitude and longitude)" ;
+ POSITION_QC:Conventions = "q where q =[0,9]" ;
+ POSITION_QC:_Fillvalue = 0 ;
+ int POSITION_QC_FLAGS(N_OBS, N_QCF) ;
+ POSITION_QC_FLAGS:long_name = "Quality flags on position" ;
+ POSITION_QC_FLAGS:Conventions = "NEMOVAR flag conventions" ;
+ POSITION_QC_FLAGS:_Fillvalue = 0 ;
+ int JULD_QC(N_OBS) ;
+ JULD_QC:long_name = "Quality on date and time" ;
+ JULD_QC:Conventions = "q where q =[0,9]" ;
+ JULD_QC:_Fillvalue = 0 ;
+ int JULD_QC_FLAGS(N_OBS, N_QCF) ;
+ JULD_QC_FLAGS:long_name = "Quality flags on date and time" ;
+ JULD_QC_FLAGS:Conventions = "NEMOVAR flag conventions" ;
+ JULD_QC_FLAGS:_Fillvalue = 0 ;
+ int ORIGINAL_FILE_INDEX(N_OBS) ;
+ ORIGINAL_FILE_INDEX:long_name = "Index in original data file" ;
+ ORIGINAL_FILE_INDEX:_Fillvalue = -99999 ;
+ float POTM_OBS(N_OBS, N_LEVELS) ;
+ POTM_OBS:long_name = "Potential temperature" ;
+ POTM_OBS:units = "Degrees Celsius" ;
+ POTM_OBS:_Fillvalue = 99999.f ;
+ float POTM_Hx(N_OBS, N_LEVELS) ;
+ POTM_Hx:long_name = "Model interpolated potential temperature" ;
+ POTM_Hx:units = "Degrees Celsius" ;
+ POTM_Hx:_Fillvalue = 99999.f ;
+ int POTM_QC(N_OBS) ;
+ POTM_QC:long_name = "Quality on potential temperature" ;
+ POTM_QC:Conventions = "q where q =[0,9]" ;
+ POTM_QC:_Fillvalue = 0 ;
+ int POTM_QC_FLAGS(N_OBS, N_QCF) ;
+ POTM_QC_FLAGS:long_name = "Quality flags on potential temperature" ;
+ POTM_QC_FLAGS:Conventions = "NEMOVAR flag conventions" ;
+ POTM_QC_FLAGS:_Fillvalue = 0 ;
+ int POTM_LEVEL_QC(N_OBS, N_LEVELS) ;
+ POTM_LEVEL_QC:long_name = "Quality for each level on potential temperature" ;
+ POTM_LEVEL_QC:Conventions = "q where q =[0,9]" ;
+ POTM_LEVEL_QC:_Fillvalue = 0 ;
+ int POTM_LEVEL_QC_FLAGS(N_OBS, N_LEVELS, N_QCF) ;
+ POTM_LEVEL_QC_FLAGS:long_name = "Quality flags for each level on potential temperature" ;
+ POTM_LEVEL_QC_FLAGS:Conventions = "NEMOVAR flag conventions" ;
+ POTM_LEVEL_QC_FLAGS:_Fillvalue = 0 ;
+ int POTM_IOBSI(N_OBS) ;
+ POTM_IOBSI:long_name = "ORCA grid search I coordinate" ;
+ int POTM_IOBSJ(N_OBS) ;
+ POTM_IOBSJ:long_name = "ORCA grid search J coordinate" ;
+ int POTM_IOBSK(N_OBS, N_LEVELS) ;
+ POTM_IOBSK:long_name = "ORCA grid search K coordinate" ;
+ char POTM_GRID(STRINGGRID) ;
+ POTM_GRID:long_name = "ORCA grid search grid (T,U,V)" ;
+ float PSAL_OBS(N_OBS, N_LEVELS) ;
+ PSAL_OBS:long_name = "Practical salinity" ;
+ PSAL_OBS:units = "PSU" ;
+ PSAL_OBS:_Fillvalue = 99999.f ;
+ float PSAL_Hx(N_OBS, N_LEVELS) ;
+ PSAL_Hx:long_name = "Model interpolated practical salinity" ;
+ PSAL_Hx:units = "PSU" ;
+ PSAL_Hx:_Fillvalue = 99999.f ;
+ int PSAL_QC(N_OBS) ;
+ PSAL_QC:long_name = "Quality on practical salinity" ;
+ PSAL_QC:Conventions = "q where q =[0,9]" ;
+ PSAL_QC:_Fillvalue = 0 ;
+ int PSAL_QC_FLAGS(N_OBS, N_QCF) ;
+ PSAL_QC_FLAGS:long_name = "Quality flags on practical salinity" ;
+ PSAL_QC_FLAGS:Conventions = "NEMOVAR flag conventions" ;
+ PSAL_QC_FLAGS:_Fillvalue = 0 ;
+ int PSAL_LEVEL_QC(N_OBS, N_LEVELS) ;
+ PSAL_LEVEL_QC:long_name = "Quality for each level on practical salinity" ;
+ PSAL_LEVEL_QC:Conventions = "q where q =[0,9]" ;
+ PSAL_LEVEL_QC:_Fillvalue = 0 ;
+ int PSAL_LEVEL_QC_FLAGS(N_OBS, N_LEVELS, N_QCF) ;
+ PSAL_LEVEL_QC_FLAGS:long_name = "Quality flags for each level on practical salinity" ;
+ PSAL_LEVEL_QC_FLAGS:Conventions = "NEMOVAR flag conventions" ;
+ PSAL_LEVEL_QC_FLAGS:_Fillvalue = 0 ;
+ int PSAL_IOBSI(N_OBS) ;
+ PSAL_IOBSI:long_name = "ORCA grid search I coordinate" ;
+ int PSAL_IOBSJ(N_OBS) ;
+ PSAL_IOBSJ:long_name = "ORCA grid search J coordinate" ;
+ int PSAL_IOBSK(N_OBS, N_LEVELS) ;
+ PSAL_IOBSK:long_name = "ORCA grid search K coordinate" ;
+ char PSAL_GRID(STRINGGRID) ;
+ PSAL_GRID:long_name = "ORCA grid search grid (T,U,V)" ;
+ float TEMP(N_OBS, N_LEVELS) ;
+ TEMP:long_name = "Insitu temperature" ;
+ TEMP:units = "Degrees Celsius" ;
+ TEMP:_Fillvalue = 99999.f ;
+
+// global attributes:
+ :title = "NEMO observation operator output" ;
+ :Convention = "NEMO unified observation operator output" ;
+}
+\end{verbatim}
+\end{alltt}
+
+\subsection{Sea level anomaly feedback type observation file header}
+
+\begin{alltt}
+\tiny
+\begin{verbatim}
+netcdf sla_01 {
+dimensions:
+ N_OBS = 41301 ;
+ N_LEVELS = 1 ;
+ N_VARS = 1 ;
+ N_QCF = 2 ;
+ N_ENTRIES = 1 ;
+ N_EXTRA = 1 ;
+ STRINGNAM = 8 ;
+ STRINGGRID = 1 ;
+ STRINGWMO = 8 ;
+ STRINGTYP = 4 ;
+ STRINGJULD = 14 ;
+variables:
+ char VARIABLES(N_VARS, STRINGNAM) ;
+ VARIABLES:long_name = "List of variables in feedback files" ;
+ char ENTRIES(N_ENTRIES, STRINGNAM) ;
+ ENTRIES:long_name = "List of additional entries for each variable in feedback files" ;
+ char EXTRA(N_EXTRA, STRINGNAM) ;
+ EXTRA:long_name = "List of extra variables" ;
+ char STATION_IDENTIFIER(N_OBS, STRINGWMO) ;
+ STATION_IDENTIFIER:long_name = "Station identifier" ;
+ char STATION_TYPE(N_OBS, STRINGTYP) ;
+ STATION_TYPE:long_name = "Code instrument type" ;
+ double LONGITUDE(N_OBS) ;
+ LONGITUDE:long_name = "Longitude" ;
+ LONGITUDE:units = "degrees_east" ;
+ LONGITUDE:_Fillvalue = 99999.f ;
+ double LATITUDE(N_OBS) ;
+ LATITUDE:long_name = "Latitude" ;
+ LATITUDE:units = "degrees_north" ;
+ LATITUDE:_Fillvalue = 99999.f ;
+ double DEPTH(N_OBS, N_LEVELS) ;
+ DEPTH:long_name = "Depth" ;
+ DEPTH:units = "metre" ;
+ DEPTH:_Fillvalue = 99999.f ;
+ int DEPTH_QC(N_OBS, N_LEVELS) ;
+ DEPTH_QC:long_name = "Quality on depth" ;
+ DEPTH_QC:Conventions = "q where q =[0,9]" ;
+ DEPTH_QC:_Fillvalue = 0 ;
+ int DEPTH_QC_FLAGS(N_OBS, N_LEVELS, N_QCF) ;
+ DEPTH_QC_FLAGS:long_name = "Quality flags on depth" ;
+ DEPTH_QC_FLAGS:Conventions = "NEMOVAR flag conventions" ;
+ double JULD(N_OBS) ;
+ JULD:long_name = "Julian day" ;
+ JULD:units = "days since JULD_REFERENCE" ;
+ JULD:Conventions = "relative julian days with decimal part (as parts of day)" ;
+ JULD:_Fillvalue = 99999.f ;
+ char JULD_REFERENCE(STRINGJULD) ;
+ JULD_REFERENCE:long_name = "Date of reference for julian days" ;
+ JULD_REFERENCE:Conventions = "YYYYMMDDHHMMSS" ;
+ int OBSERVATION_QC(N_OBS) ;
+ OBSERVATION_QC:long_name = "Quality on observation" ;
+ OBSERVATION_QC:Conventions = "q where q =[0,9]" ;
+ OBSERVATION_QC:_Fillvalue = 0 ;
+ int OBSERVATION_QC_FLAGS(N_OBS, N_QCF) ;
+ OBSERVATION_QC_FLAGS:long_name = "Quality flags on observation" ;
+ OBSERVATION_QC_FLAGS:Conventions = "NEMOVAR flag conventions" ;
+ OBSERVATION_QC_FLAGS:_Fillvalue = 0 ;
+ int POSITION_QC(N_OBS) ;
+ POSITION_QC:long_name = "Quality on position (latitude and longitude)" ;
+ POSITION_QC:Conventions = "q where q =[0,9]" ;
+ POSITION_QC:_Fillvalue = 0 ;
+ int POSITION_QC_FLAGS(N_OBS, N_QCF) ;
+ POSITION_QC_FLAGS:long_name = "Quality flags on position" ;
+ POSITION_QC_FLAGS:Conventions = "NEMOVAR flag conventions" ;
+ POSITION_QC_FLAGS:_Fillvalue = 0 ;
+ int JULD_QC(N_OBS) ;
+ JULD_QC:long_name = "Quality on date and time" ;
+ JULD_QC:Conventions = "q where q =[0,9]" ;
+ JULD_QC:_Fillvalue = 0 ;
+ int JULD_QC_FLAGS(N_OBS, N_QCF) ;
+ JULD_QC_FLAGS:long_name = "Quality flags on date and time" ;
+ JULD_QC_FLAGS:Conventions = "NEMOVAR flag conventions" ;
+ JULD_QC_FLAGS:_Fillvalue = 0 ;
+ int ORIGINAL_FILE_INDEX(N_OBS) ;
+ ORIGINAL_FILE_INDEX:long_name = "Index in original data file" ;
+ ORIGINAL_FILE_INDEX:_Fillvalue = -99999 ;
+ float SLA_OBS(N_OBS, N_LEVELS) ;
+ SLA_OBS:long_name = "Sea level anomaly" ;
+ SLA_OBS:units = "metre" ;
+ SLA_OBS:_Fillvalue = 99999.f ;
+ float SLA_Hx(N_OBS, N_LEVELS) ;
+ SLA_Hx:long_name = "Model interpolated sea level anomaly" ;
+ SLA_Hx:units = "metre" ;
+ SLA_Hx:_Fillvalue = 99999.f ;
+ int SLA_QC(N_OBS) ;
+ SLA_QC:long_name = "Quality on sea level anomaly" ;
+ SLA_QC:Conventions = "q where q =[0,9]" ;
+ SLA_QC:_Fillvalue = 0 ;
+ int SLA_QC_FLAGS(N_OBS, N_QCF) ;
+ SLA_QC_FLAGS:long_name = "Quality flags on sea level anomaly" ;
+ SLA_QC_FLAGS:Conventions = "NEMOVAR flag conventions" ;
+ SLA_QC_FLAGS:_Fillvalue = 0 ;
+ int SLA_LEVEL_QC(N_OBS, N_LEVELS) ;
+ SLA_LEVEL_QC:long_name = "Quality for each level on sea level anomaly" ;
+ SLA_LEVEL_QC:Conventions = "q where q =[0,9]" ;
+ SLA_LEVEL_QC:_Fillvalue = 0 ;
+ int SLA_LEVEL_QC_FLAGS(N_OBS, N_LEVELS, N_QCF) ;
+ SLA_LEVEL_QC_FLAGS:long_name = "Quality flags for each level on sea level anomaly" ;
+ SLA_LEVEL_QC_FLAGS:Conventions = "NEMOVAR flag conventions" ;
+ SLA_LEVEL_QC_FLAGS:_Fillvalue = 0 ;
+ int SLA_IOBSI(N_OBS) ;
+ SLA_IOBSI:long_name = "ORCA grid search I coordinate" ;
+ int SLA_IOBSJ(N_OBS) ;
+ SLA_IOBSJ:long_name = "ORCA grid search J coordinate" ;
+ int SLA_IOBSK(N_OBS, N_LEVELS) ;
+ SLA_IOBSK:long_name = "ORCA grid search K coordinate" ;
+ char SLA_GRID(STRINGGRID) ;
+ SLA_GRID:long_name = "ORCA grid search grid (T,U,V)" ;
+ float MDT(N_OBS, N_LEVELS) ;
+ MDT:long_name = "Mean Dynamic Topography" ;
+ MDT:units = "metre" ;
+ MDT:_Fillvalue = 99999.f ;
+
+// global attributes:
+ :title = "NEMO observation operator output" ;
+ :Convention = "NEMO unified observation operator output" ;
+}
+\end{verbatim}
+\end{alltt}
+
+The mean dynamic
+topography (MDT) must be provided in a separate file defined on the model grid
+ called {\it slaReferenceLevel.nc}. The MDT is required in
+order to produce the model equivalent sea level anomaly from the model sea
+surface height. Below is an example header for this file (on the ORCA025 grid).
+
+\begin{alltt}
+\tiny
+\begin{verbatim}
+dimensions:
+ x = 1442 ;
+ y = 1021 ;
+variables:
+ float nav_lon(y, x) ;
+ nav_lon:units = "degrees_east" ;
+ float nav_lat(y, x) ;
+ nav_lat:units = "degrees_north" ;
+ float sossheig(y, x) ;
+ sossheig:_FillValue = -1.e+30f ;
+ sossheig:coordinates = "nav_lon nav_lat" ;
+ sossheig:long_name = "Mean Dynamic Topography" ;
+ sossheig:units = "metres" ;
+ sossheig:grid = "orca025T" ;
+\end{verbatim}
+\end{alltt}
+
+\subsection{Sea surface temperature feedback type observation file header}
+
+\begin{alltt}
+\tiny
+\begin{verbatim}
+netcdf sst_01 {
+dimensions:
+ N_OBS = 33099 ;
+ N_LEVELS = 1 ;
+ N_VARS = 1 ;
+ N_QCF = 2 ;
+ N_ENTRIES = 1 ;
+ STRINGNAM = 8 ;
+ STRINGGRID = 1 ;
+ STRINGWMO = 8 ;
+ STRINGTYP = 4 ;
+ STRINGJULD = 14 ;
+variables:
+ char VARIABLES(N_VARS, STRINGNAM) ;
+ VARIABLES:long_name = "List of variables in feedback files" ;
+ char ENTRIES(N_ENTRIES, STRINGNAM) ;
+ ENTRIES:long_name = "List of additional entries for each variable in feedback files" ;
+ char STATION_IDENTIFIER(N_OBS, STRINGWMO) ;
+ STATION_IDENTIFIER:long_name = "Station identifier" ;
+ char STATION_TYPE(N_OBS, STRINGTYP) ;
+ STATION_TYPE:long_name = "Code instrument type" ;
+ double LONGITUDE(N_OBS) ;
+ LONGITUDE:long_name = "Longitude" ;
+ LONGITUDE:units = "degrees_east" ;
+ LONGITUDE:_Fillvalue = 99999.f ;
+ double LATITUDE(N_OBS) ;
+ LATITUDE:long_name = "Latitude" ;
+ LATITUDE:units = "degrees_north" ;
+ LATITUDE:_Fillvalue = 99999.f ;
+ double DEPTH(N_OBS, N_LEVELS) ;
+ DEPTH:long_name = "Depth" ;
+ DEPTH:units = "metre" ;
+ DEPTH:_Fillvalue = 99999.f ;
+ int DEPTH_QC(N_OBS, N_LEVELS) ;
+ DEPTH_QC:long_name = "Quality on depth" ;
+ DEPTH_QC:Conventions = "q where q =[0,9]" ;
+ DEPTH_QC:_Fillvalue = 0 ;
+ int DEPTH_QC_FLAGS(N_OBS, N_LEVELS, N_QCF) ;
+ DEPTH_QC_FLAGS:long_name = "Quality flags on depth" ;
+ DEPTH_QC_FLAGS:Conventions = "NEMOVAR flag conventions" ;
+ double JULD(N_OBS) ;
+ JULD:long_name = "Julian day" ;
+ JULD:units = "days since JULD_REFERENCE" ;
+ JULD:Conventions = "relative julian days with decimal part (as parts of day)" ;
+ JULD:_Fillvalue = 99999.f ;
+ char JULD_REFERENCE(STRINGJULD) ;
+ JULD_REFERENCE:long_name = "Date of reference for julian days" ;
+ JULD_REFERENCE:Conventions = "YYYYMMDDHHMMSS" ;
+ int OBSERVATION_QC(N_OBS) ;
+ OBSERVATION_QC:long_name = "Quality on observation" ;
+ OBSERVATION_QC:Conventions = "q where q =[0,9]" ;
+ OBSERVATION_QC:_Fillvalue = 0 ;
+ int OBSERVATION_QC_FLAGS(N_OBS, N_QCF) ;
+ OBSERVATION_QC_FLAGS:long_name = "Quality flags on observation" ;
+ OBSERVATION_QC_FLAGS:Conventions = "NEMOVAR flag conventions" ;
+ OBSERVATION_QC_FLAGS:_Fillvalue = 0 ;
+ int POSITION_QC(N_OBS) ;
+ POSITION_QC:long_name = "Quality on position (latitude and longitude)" ;
+ POSITION_QC:Conventions = "q where q =[0,9]" ;
+ POSITION_QC:_Fillvalue = 0 ;
+ int POSITION_QC_FLAGS(N_OBS, N_QCF) ;
+ POSITION_QC_FLAGS:long_name = "Quality flags on position" ;
+ POSITION_QC_FLAGS:Conventions = "NEMOVAR flag conventions" ;
+ POSITION_QC_FLAGS:_Fillvalue = 0 ;
+ int JULD_QC(N_OBS) ;
+ JULD_QC:long_name = "Quality on date and time" ;
+ JULD_QC:Conventions = "q where q =[0,9]" ;
+ JULD_QC:_Fillvalue = 0 ;
+ int JULD_QC_FLAGS(N_OBS, N_QCF) ;
+ JULD_QC_FLAGS:long_name = "Quality flags on date and time" ;
+ JULD_QC_FLAGS:Conventions = "NEMOVAR flag conventions" ;
+ JULD_QC_FLAGS:_Fillvalue = 0 ;
+ int ORIGINAL_FILE_INDEX(N_OBS) ;
+ ORIGINAL_FILE_INDEX:long_name = "Index in original data file" ;
+ ORIGINAL_FILE_INDEX:_Fillvalue = -99999 ;
+ float SST_OBS(N_OBS, N_LEVELS) ;
+ SST_OBS:long_name = "Sea surface temperature" ;
+ SST_OBS:units = "Degree centigrade" ;
+ SST_OBS:_Fillvalue = 99999.f ;
+ float SST_Hx(N_OBS, N_LEVELS) ;
+ SST_Hx:long_name = "Model interpolated sea surface temperature" ;
+ SST_Hx:units = "Degree centigrade" ;
+ SST_Hx:_Fillvalue = 99999.f ;
+ int SST_QC(N_OBS) ;
+ SST_QC:long_name = "Quality on sea surface temperature" ;
+ SST_QC:Conventions = "q where q =[0,9]" ;
+ SST_QC:_Fillvalue = 0 ;
+ int SST_QC_FLAGS(N_OBS, N_QCF) ;
+ SST_QC_FLAGS:long_name = "Quality flags on sea surface temperature" ;
+ SST_QC_FLAGS:Conventions = "NEMOVAR flag conventions" ;
+ SST_QC_FLAGS:_Fillvalue = 0 ;
+ int SST_LEVEL_QC(N_OBS, N_LEVELS) ;
+ SST_LEVEL_QC:long_name = "Quality for each level on sea surface temperature" ;
+ SST_LEVEL_QC:Conventions = "q where q =[0,9]" ;
+ SST_LEVEL_QC:_Fillvalue = 0 ;
+ int SST_LEVEL_QC_FLAGS(N_OBS, N_LEVELS, N_QCF) ;
+ SST_LEVEL_QC_FLAGS:long_name = "Quality flags for each level on sea surface temperature" ;
+ SST_LEVEL_QC_FLAGS:Conventions = "NEMOVAR flag conventions" ;
+ SST_LEVEL_QC_FLAGS:_Fillvalue = 0 ;
+ int SST_IOBSI(N_OBS) ;
+ SST_IOBSI:long_name = "ORCA grid search I coordinate" ;
+ int SST_IOBSJ(N_OBS) ;
+ SST_IOBSJ:long_name = "ORCA grid search J coordinate" ;
+ int SST_IOBSK(N_OBS, N_LEVELS) ;
+ SST_IOBSK:long_name = "ORCA grid search K coordinate" ;
+ char SST_GRID(STRINGGRID) ;
+ SST_GRID:long_name = "ORCA grid search grid (T,U,V)" ;
+
+// global attributes:
+ :title = "NEMO observation operator output" ;
+ :Convention = "NEMO unified observation operator output" ;
+}
+\end{verbatim}
+\end{alltt}
+
+\section{Theoretical details}
+\label{OBS_theory}
+
+\subsection{Horizontal interpolation methods}
+
+Consider an observation point ${\rm P}$ with
+with longitude and latitude $({\lambda_{}}_{\rm P}, \phi_{\rm P})$ and the
+four nearest neighbouring model grid points ${\rm A}$, ${\rm B}$, ${\rm C}$
+and ${\rm D}$ with longitude and latitude ($\lambda_{\rm A}$, $\phi_{\rm A}$),
+($\lambda_{\rm B}$, $\phi_{\rm B}$) etc.
+All horizontal interpolation methods implemented in NEMO
+estimate the value of a model variable $x$ at point $P$ as
+a weighted linear combination of the values of the model
+variables at the grid points ${\rm A}$, ${\rm B}$ etc.:
+\begin{eqnarray}
+{x_{}}_{\rm P} & \hspace{-2mm} = \hspace{-2mm} &
+\frac{1}{w} \left( {w_{}}_{\rm A} {x_{}}_{\rm A} +
+ {w_{}}_{\rm B} {x_{}}_{\rm B} +
+ {w_{}}_{\rm C} {x_{}}_{\rm C} +
+ {w_{}}_{\rm D} {x_{}}_{\rm D} \right)
+\end{eqnarray}
+where ${w_{}}_{\rm A}$, ${w_{}}_{\rm B}$ etc. are the respective weights for the
+model field at points ${\rm A}$, ${\rm B}$ etc., and
+$w = {w_{}}_{\rm A} + {w_{}}_{\rm B} + {w_{}}_{\rm C} + {w_{}}_{\rm D}$.
+
+Four different possibilities are available for computing the weights.
+
+\begin{enumerate}
+
+\item[1.] {\bf Great-Circle distance-weighted interpolation.} The weights
+ are computed as a function of the great-circle distance $s(P, \cdot)$
+ between $P$ and the model grid points $A$, $B$ etc. For example,
+ the weight given to the field ${x_{}}_{\rm A}$ is specified as the
+ product of the distances from ${\rm P}$ to the other points:
+ \begin{eqnarray}
+ {w_{}}_{\rm A} = s({\rm P}, {\rm B}) \, s({\rm P}, {\rm C}) \, s({\rm P}, {\rm D})
+ \nonumber
+ \end{eqnarray}
+ where
+ \begin{eqnarray}
+ s\left ({\rm P}, {\rm M} \right )
+ & \hspace{-2mm} = \hspace{-2mm} &
+ \cos^{-1} \! \left\{
+ \sin {\phi_{}}_{\rm P} \sin {\phi_{}}_{\rm M}
+ + \cos {\phi_{}}_{\rm P} \cos {\phi_{}}_{\rm M}
+ \cos ({\lambda_{}}_{\rm M} - {\lambda_{}}_{\rm P})
+ \right\}
+ \end{eqnarray}
+ and $M$ corresponds to $B$, $C$ or $D$.
+ A more stable form of the great-circle distance formula for
+ small distances ($x$ near 1) involves the arcsine function
+ ($e.g.$ see p.~101 of \citet{Daley_Barker_Bk01}:
+ \begin{eqnarray}
+ s\left( {\rm P}, {\rm M} \right)
+ & \hspace{-2mm} = \hspace{-2mm} &
+ \sin^{-1} \! \left\{ \sqrt{ 1 - x^2 } \right\}
+ \nonumber
+ \end{eqnarray}
+ where
+ \begin{eqnarray}
+ x & \hspace{-2mm} = \hspace{-2mm} &
+ {a_{}}_{\rm M} {a_{}}_{\rm P} + {b_{}}_{\rm M} {b_{}}_{\rm P} + {c_{}}_{\rm M} {c_{}}_{\rm P}
+ \nonumber
+ \end{eqnarray}
+ and
+ \begin{eqnarray}
+ {a_{}}_{\rm M} & \hspace{-2mm} = \hspace{-2mm} & \sin {\phi_{}}_{\rm M},
+ \nonumber \\
+ {a_{}}_{\rm P} & \hspace{-2mm} = \hspace{-2mm} & \sin {\phi_{}}_{\rm P},
+ \nonumber \\
+ {b_{}}_{\rm M} & \hspace{-2mm} = \hspace{-2mm} & \cos {\phi_{}}_{\rm M} \cos {\phi_{}}_{\rm M},
+ \nonumber \\
+ {b_{}}_{\rm P} & \hspace{-2mm} = \hspace{-2mm} & \cos {\phi_{}}_{\rm P} \cos {\phi_{}}_{\rm P},
+ \nonumber \\
+ {c_{}}_{\rm M} & \hspace{-2mm} = \hspace{-2mm} & \cos {\phi_{}}_{\rm M} \sin {\phi_{}}_{\rm M},
+ \nonumber \\
+ {c_{}}_{\rm P} & \hspace{-2mm} = \hspace{-2mm} & \cos {\phi_{}}_{\rm P} \sin {\phi_{}}_{\rm P}.
+ \nonumber
+ \nonumber
+ \end{eqnarray}
+
+\item[2.] {\bf Great-Circle distance-weighted interpolation with small angle
+ approximation.} Similar to the previous interpolation but with the
+ distance $s$ computed as
+ \begin{eqnarray}
+ s\left( {\rm P}, {\rm M} \right)
+ & \hspace{-2mm} = \hspace{-2mm} &
+ \sqrt{ \left( {\phi_{}}_{\rm M} - {\phi_{}}_{\rm P} \right)^{2}
+ + \left( {\lambda_{}}_{\rm M} - {\lambda_{}}_{\rm P} \right)^{2}
+ \cos^{2} {\phi_{}}_{\rm M} }
+ \end{eqnarray}
+ where $M$ corresponds to $A$, $B$, $C$ or $D$.
+
+\item[3.] {\bf Bilinear interpolation for a regular spaced grid.} The
+ interpolation is split into two 1D interpolations in the longitude
+ and latitude directions, respectively.
+
+\item[4.] {\bf Bilinear remapping interpolation for a general grid.} An
+ iterative scheme that involves first mapping a quadrilateral cell
+ into a cell with coordinates (0,0), (1,0), (0,1) and (1,1). This
+ method is based on the SCRIP interpolation package \citep{Jones_1998}.
+
+\end{enumerate}
+
+\subsection{Grid search}
+
+For many grids used by the NEMO model, such as the ORCA family,
+the horizontal grid coordinates $i$ and $j$ are not simple functions
+of latitude and longitude. Therefore, it is not always straightforward
+to determine the grid points surrounding any given observational position.
+Before the interpolation can be performed, a search
+algorithm is then required to determine the corner points of
+the quadrilateral cell in which the observation is located.
+This is the most difficult and time consuming part of the
+2D interpolation procedure.
+A robust test for determining if an observation falls
+within a given quadrilateral cell is as follows. Let
+${\rm P}({\lambda_{}}_{\rm P} ,{\phi_{}}_{\rm P} )$ denote the observation point,
+and let ${\rm A}({\lambda_{}}_{\rm A} ,{\phi_{}}_{\rm A} )$,
+${\rm B}({\lambda_{}}_{\rm B} ,{\phi_{}}_{\rm B} )$,
+${\rm C}({\lambda_{}}_{\rm C} ,{\phi_{}}_{\rm C} )$
+and
+${\rm D}({\lambda_{}}_{\rm D} ,{\phi_{}}_{\rm D} )$ denote
+the bottom left, bottom right, top left and top right
+corner points of the cell, respectively.
+To determine if P is inside
+the cell, we verify that the cross-products
+\begin{eqnarray}
+\begin{array}{lllll}
+{{\bf r}_{}}_{\rm PA} \times {{\bf r}_{}}_{\rm PC}
+& = & [({\lambda_{}}_{\rm A}\; -\; {\lambda_{}}_{\rm P} )
+ ({\phi_{}}_{\rm C} \; -\; {\phi_{}}_{\rm P} )
+ - ({\lambda_{}}_{\rm C}\; -\; {\lambda_{}}_{\rm P} )
+ ({\phi_{}}_{\rm A} \; -\; {\phi_{}}_{\rm P} )] \; \widehat{\bf k} \\
+{{\bf r}_{}}_{\rm PB} \times {{\bf r}_{}}_{\rm PA}
+& = & [({\lambda_{}}_{\rm B}\; -\; {\lambda_{}}_{\rm P} )
+ ({\phi_{}}_{\rm A} \; -\; {\phi_{}}_{\rm P} )
+ - ({\lambda_{}}_{\rm A}\; -\; {\lambda_{}}_{\rm P} )
+ ({\phi_{}}_{\rm B} \; -\; {\phi_{}}_{\rm P} )] \; \widehat{\bf k} \\
+{{\bf r}_{}}_{\rm PC} \times {{\bf r}_{}}_{\rm PD}
+& = & [({\lambda_{}}_{\rm C}\; -\; {\lambda_{}}_{\rm P} )
+ ({\phi_{}}_{\rm D} \; -\; {\phi_{}}_{\rm P} )
+ - ({\lambda_{}}_{\rm D}\; -\; {\lambda_{}}_{\rm P} )
+ ({\phi_{}}_{\rm C} \; -\; {\phi_{}}_{\rm P} )] \; \widehat{\bf k} \\
+{{\bf r}_{}}_{\rm PD} \times {{\bf r}_{}}_{\rm PB}
+& = & [({\lambda_{}}_{\rm D}\; -\; {\lambda_{}}_{\rm P} )
+ ({\phi_{}}_{\rm B} \; -\; {\phi_{}}_{\rm P} )
+ - ({\lambda_{}}_{\rm B}\; -\; {\lambda_{}}_{\rm P} )
+ ({\phi_{}}_{\rm D} \; - \; {\phi_{}}_{\rm P} )] \; \widehat{\bf k} \\
+\end{array}
+\label{eq:cross}
+\end{eqnarray}
+point in the opposite direction to the unit normal
+$\widehat{\bf k}$ (i.e., that the coefficients of
+$\widehat{\bf k}$ are negative),
+where ${{\bf r}_{}}_{\rm PA}$, ${{\bf r}_{}}_{\rm PB}$,
+etc. correspond to the vectors between points P and A,
+P and B, etc.. The method used is
+similar to the method used in
+the SCRIP interpolation package \citep{Jones_1998}.
+
+In order to speed up the grid search, there is the possibility to construct
+a lookup table for a user specified resolution. This lookup
+table contains the lower and upper bounds on the $i$ and $j$ indices
+to be searched for on a regular grid. For each observation position,
+the closest point on the regular grid of this position is computed and
+the $i$ and $j$ ranges of this point searched to determine the precise
+four points surrounding the observation.
+
+\subsection{Parallel aspects of horizontal interpolation}
+\label{OBS_parallel}
+
+For horizontal interpolation, there is the basic problem that the
+observations are unevenly distributed on the globe. In numerical
+models, it is common to divide the model grid into subgrids (or
+domains) where each subgrid is executed on a single processing element
+with explicit message passing for exchange of information along the
+domain boundaries when running on a massively parallel processor (MPP)
+system. This approach is used by \NEMO.
+
+For observations there is no natural distribution since the
+observations are not equally distributed on the globe.
+Two options have been made available: 1) geographical distribution;
+and 2) round-robin.
+
+\subsubsection{Geographical distribution of observations among processors}
+
+%>>>>>>>>>>>>>>>>>>>>>>>>>>>>
+\begin{figure} \begin{center}
+\includegraphics[width=10cm,height=12cm,angle=-90.]{./TexFiles/Figures/Fig_ASM_obsdist_local}
+\caption{ \label{fig:obslocal}
+Example of the distribution of observations with the geographical distribution of observational data.}
+\end{center} \end{figure}
+%>>>>>>>>>>>>>>>>>>>>>>>>>>>>
+
+This is the simplest option in which the observations are distributed according
+to the domain of the grid-point parallelization. Figure~\ref{fig:obslocal}
+shows an example of the distribution of the {\em in situ} data on processors
+with a different colour for each observation
+on a given processor for a 4 $\times$ 2 decomposition with ORCA2.
+The grid-point domain decomposition is clearly visible on the plot.
+
+The advantage of this approach is that all
+information needed for horizontal interpolation is available without
+any MPP communication. Of course, this is under the assumption that
+we are only using a $2 \times 2$ grid-point stencil for the interpolation
+(e.g., bilinear interpolation). For higher order interpolation schemes this
+is no longer valid. A disadvantage with the above scheme is that the number of
+observations on each processor can be very different. If the cost of
+the actual interpolation is expensive relative to the communication of
+data needed for interpolation, this could lead to load imbalance.
+
+\subsubsection{Round-robin distribution of observations among processors}
+
+%>>>>>>>>>>>>>>>>>>>>>>>>>>>>
+\begin{figure} \begin{center}
+\includegraphics[width=10cm,height=12cm,angle=-90.]{./TexFiles/Figures/Fig_ASM_obsdist_global}
+\caption{ \label{fig:obsglobal}
+Example of the distribution of observations with the round-robin distribution of observational data.}
+\end{center} \end{figure}
+%>>>>>>>>>>>>>>>>>>>>>>>>>>>>
+
+An alternative approach is to distribute the observations equally
+among processors and use message passing in order to retrieve
+the stencil for interpolation. The simplest distribution of the observations
+is to distribute them using a round-robin scheme. Figure~\ref{fig:obsglobal}
+shows the distribution of the {\em in situ} data on processors for the
+round-robin distribution of observations with a different colour for
+each observation on a given processor for a 4 $\times$ 2 decomposition
+with ORCA2 for the same input data as in Fig.~\ref{fig:obslocal}.
+The observations are now clearly randomly distributed on the globe.
+In order to be able to perform horizontal interpolation in this case,
+a subroutine has been developed that retrieves any grid points in the
+global space.
+
+\subsection{Vertical interpolation operator}
+
+Vertical interpolation is achieved using either a cubic spline or
+linear interpolation. For the cubic spline, the top and
+bottom boundary conditions for the second derivative of the
+interpolating polynomial in the spline are set to zero.
+At the bottom boundary, this is done using the land-ocean mask.
+
+\newpage
+
+% ================================================================
+% Offline observation operator documentation
+% ================================================================
+
+%\usepackage{framed}
+
+\section{Offline observation operator}
+\label{OBS_ooo}
+
+\subsection{Concept}
+
+The obs oper maps model variables to observation space. It is possible to apply this mapping
+without running the model. The software which performs this functionality is known as the
+\textbf{offline obs oper}. The obs oper is divided into three stages. An initialisation phase,
+an interpolation phase and an output phase. The implementation of which is outlined in the
+previous sections. During the interpolation phase the offline obs oper populates the model
+arrays by reading saved model fields from disk.
+
+There are two ways of exploiting this offline capacity. The first is to mimic the behaviour of
+the online system by supplying model fields at regular intervals between the start and the end
+of the run. This approach results in a single model counterpart per observation. This kind of
+usage produces feedback files the same file format as the online obs oper.
+The second is to take advantage of the offline setting in which multiple model counterparts can
+be calculated per observation. In this case it is possible to consider all forecasts verifying
+at the same time. By forecast, I mean any method which produces an estimate of physical reality
+which is not an observed value. In the case of class 4 files this means forecasts, analyses, persisted
+analyses and climatological values verifying at the same time. Although the class 4 file format
+doesn't account for multiple ensemble members or multiple experiments per observation, it is possible
+to include these components in the same or multiple files.
+
+%--------------------------------------------------------------------------------------------------------
+% offline_oper.exe
+%--------------------------------------------------------------------------------------------------------
+
+\subsection{Using the offline observation operator}
+
+\subsubsection{Building}
+
+In addition to \emph{OPA\_SRC} the offline obs oper requires the inclusion
+of the \emph{OOO\_SRC} directory. \emph{OOO\_SRC} contains a replacement \textbf{nemo.f90} and
+\textbf{nemogcm.F90} which overwrites the resultant \textbf{nemo.exe}. This is the approach taken
+by \emph{SAS\_SRC} and \emph{OFF\_SRC}.
+
+%--------------------------------------------------------------------------------------------------------
+% Running
+%--------------------------------------------------------------------------------------------------------
+\subsubsection{Running}
+
+The simplest way to use the executable is to edit and append the \textbf{ooo.nml} namelist to
+a full NEMO namelist and then to run the executable as if it were nemo.exe.
+
+\subsubsection{Quick script}
+
+A useful Python utility to control the namelist options can be found in \textbf{OBSTOOLS/OOO}. The
+functions which locate model fields and observation files can be manually specified. The package
+can be installed by appropriate use of the included setup.py script.
+
+Documentation can be auto-generated by Sphinx by running \emph{make html} in the \textbf{doc} directory.
+
+%--------------------------------------------------------------------------------------------------------
+% Configuration section
+%--------------------------------------------------------------------------------------------------------
+\subsection{Configuring the offline observation operator}
+The observation files and settings understood by \textbf{namobs} have been outlined in the online
+obs oper section. In addition there are two further namelists wich control the operation of the offline
+obs oper. \textbf{namooo} which controls the input model fields and \textbf{namcl4} which controls the
+production of class 4 files.
+
+\subsubsection{Single field}
+
+In offline mode model arrays are populated at appropriate time steps via input files.
+At present, \textbf{tsn} and \textbf{sshn} are populated by the default read routines.
+These routines will be expanded upon in future versions to allow the specification of any
+model variable. As such, input files must be global versions of the model domain with
+\textbf{votemper}, \textbf{vosaline} and optionally \textbf{sshn} present.
+
+For each field read there must be an entry in the \textbf{namooo} namelist specifying the
+name of the file to read and the index along the \emph{time\_counter}. For example, to
+read the second time counter from a single file the namelist would be.
+
+\begin{alltt}
+\tiny
+\begin{verbatim}
+!----------------------------------------------------------------------
+! namooo Offline obs_oper namelist
+!----------------------------------------------------------------------
+! ooo_files specifies the files containing the model counterpart
+! nn_ooo_idx specifies the time_counter index within the model file
+&namooo
+ ooo_files = "foo.nc"
+ nn_ooo_idx = 2
+/
+\end{verbatim}
+\end{alltt}
+
+\subsubsection{Multiple fields per run}
+
+Model field iteration is controlled via \textbf{nn\_ooo\_freq} which specifies
+the number of model steps at which the next field gets read. For example, if
+12 hourly fields are to be interpolated in a setup where 288 steps equals 24 hours.
+
+\begin{alltt}
+\tiny
+\begin{verbatim}
+!----------------------------------------------------------------------
+! namooo Offline obs_oper namelist
+!----------------------------------------------------------------------
+! ooo_files specifies the files containing the model counterpart
+! nn_ooo_idx specifies the time_counter index within the model file
+! nn_ooo_freq specifies number of time steps between read operations
+&namooo
+ ooo_files = "foo.nc" "foo.nc"
+ nn_ooo_idx = 1 2
+ nn_ooo_freq = 144
+/
+\end{verbatim}
+\end{alltt}
+
+The above namelist will result in feedback files whose first 12 hours contain
+the first field of foo.nc and the second 12 hours contain the second field.
+
+%\begin{framed}
+\textbf{Note} Missing files can be denoted as "nofile".
+%\end{framed}
+
+It is easy to see how a collection of fields taken fron a number of files
+at different indices can be combined at a particular frequency in time to
+generate a pseudo model evolution. As long as all that is needed is a single
+model counterpart at a regular interval then namooo is all that needs to
+be edited. However, a far more interesting approach can be taken in which
+multiple forecasts, analyses, persisted analyses and climatologies are
+considered against the same set of observations. For this a slightly more
+complicated approach is needed. It is referred to as \emph{Class 4} since
+it is the fourth metric defined by the GODAE intercomparison project.
+
+%--------------------------------------------------------------------------------------------------------
+% Class 4 file section
+%--------------------------------------------------------------------------------------------------------
+\subsubsection{Multiple model counterparts per observation a.k.a Class 4}
+
+A generalisation of feedback files to allow multiple model components per observation. For a single
+observation, as well as previous forecasts verifying at the same time there are also analyses, persisted
+analyses and climatologies.
+
+
+The above namelist performs two basic functions. It organises the fields
+given in \textbf{namooo} into groups so that observations can be matched
+up multiple times. It also controls the metadata and the output variable
+of the class 4 file when a write routine is called.
+
+%\begin{framed}
+\textbf{Note: ln\_cl4} must be set to \emph{.TRUE.} in \textbf{namobs}
+to use class 4 outputs.
+%\end{framed}
+
+\subsubsection{Class 4 naming convention}
+
+The standard class 4 file naming convention is as follows.
+
+\noindent
+\linebreak
+\textbf{\$\{prefix\}\_\$\{yyyymmdd\}\_\$\{sys\}\_\$\{cfg\}\_\$\{vn\}\_\$\{kind\}\_\$\{nproc\}.nc}
+
+\noindent
+\linebreak
+Much of the namelist is devoted to specifying this convention. The
+following namelist settings control the elements of the output
+file names. Each should be specified as a single string of character data.
+
+\begin{description}
+\item[cl4\_prefix]
+Prefix for class 4 files e.g. class4
+\item[cl4\_date]
+YYYYMMDD validity date
+\item[cl4\_sys]
+The name of the class 4 model system e.g. FOAM
+\item[cl4\_cfg]
+The name of the class 4 model configuration e.g. orca025
+\item[cl4\_vn]
+The name of the class 4 model version e.g. 12.0
+\end{description}
+
+\noindent
+The kind is specified by the observation type internally to the obs oper. The processor
+number is specified internally in NEMO.
+
+\subsubsection{Class 4 file global attributes}
+
+Global attributes necessary to fulfill the class 4 file definition. These
+are also useful pieces of information when collaborating with external
+partners.
+
+\begin{description}
+\item[cl4\_contact]
+Contact email for class 4 files.
+\item[cl4\_inst]
+The name of the producers institution.
+\item[cl4\_cfg]
+The name of the class 4 model configuration e.g. orca025
+\item[cl4\_vn]
+The name of the class 4 model version e.g. 12.0
+\end{description}
+
+\noindent
+The obs\_type,
+creation date and validity time are specified internally to the obs oper.
+
+\subsubsection{Class 4 model counterpart configuration}
+
+As seen previously it is possible to perform a single sweep of the
+obs oper and specify a collection of model fields equally spaced
+along that sweep. In the class 4 case the single sweep is replaced
+with multiple sweeps and a certain ammount of book keeping is
+needed to ensure each model counterpart makes its way to the
+correct piece of memory in the output files.
+
+\noindent
+\linebreak
+In terms of book keeping, the offline obs oper needs to know how many
+full sweeps need to be performed. This is specified via the
+\textbf{cl4\_match\_len} variable and is the total number of model
+counterparts per observation. For example, a 3 forecasts plus 3 persistence
+fields plus an analysis field would be 7 counterparts per observation.
+
+\begin{alltt}
+\tiny
+\begin{verbatim}
+ cl4_match_len = 7
+\end{verbatim}
+\end{alltt}
+
+Then to correctly allocate a class 4 file the forecast axis must be defined. This
+is controlled via \textbf{cl4\_fcst\_len}, which in out above example would be 3.
+
+\begin{alltt}
+\tiny
+\begin{verbatim}
+ cl4_fcst_len = 3
+\end{verbatim}
+\end{alltt}
+
+Then for each model field it is necessary to designate what class 4 variable and
+index along the forecast dimension the model counterpart should be stored in the
+output file. As well as a value for that lead time in hours, this will be useful
+when interpreting the data afterwards.
+
+\begin{alltt}
+\tiny
+\begin{verbatim}
+ cl4_vars = "forecast" "forecast" "forecast" "persistence" "persistence"
+ "persistence" "best_estimate"
+ cl4_fcst_idx = 1 2 3 1 2 3 1
+ cl4_leadtime = 12 36 60
+\end{verbatim}
+\end{alltt}
+
+In terms of files and indices of fields inside each file the class 4 approach
+makes use of the \textbf{namooo} namelist. If our fields are in separate files
+with a single field per file our example inputs will be specified.
+
+\begin{alltt}
+\tiny
+\begin{verbatim}
+ ooo_files = "F.1.nc" "F.2.nc" "F.3.nc" "P.1.nc" "P.2.nc" "P.3.nc" "A.1.nc"
+ nn_ooo_idx = 1 1 1 1 1 1 1
+\end{verbatim}
+\end{alltt}
+
+When we combine all of the naming conventions, global attributes and i/o instructions
+the class 4 namelist becomes.
+
+\begin{alltt}
+\tiny
+\begin{verbatim}
+!----------------------------------------------------------------------
+! namooo Offline obs_oper namelist
+!----------------------------------------------------------------------
+! ooo_files specifies the files containing the model counterpart
+! nn_ooo_idx specifies the time_counter index within the model file
+! nn_ooo_freq specifies number of time steps between read operations
+&namooo
+ ooo_files = "F.1.nc" "F.2.nc" "F.3.nc" "P.1.nc" "P.2.nc" "P.3.nc" "A.1.nc"
+ nn_ooo_idx = 1 1 1 1 1 1 1
+/
+!----------------------------------------------------------------------
+! namcl4 Offline obs_oper class 4 namelist
+!----------------------------------------------------------------------
+!
+! Naming convention
+! -----------------
+! cl4_prefix specifies the output file prefix
+! cl4_date specifies the output file validity date
+! cl4_sys specifies the model counterpart system
+! cl4_cfg specifies the model counterpart configuration
+! cl4_vn specifies the model counterpart version
+! cl4_inst specifies the model counterpart institute
+! cl4_contact specifies the file producers contact details
+!
+! I/O specification
+! -----------------
+! cl4_vars specifies the names of the output file netcdf variable
+! cl4_fcst_idx specifies output file forecast index
+! cl4_fcst_len specifies forecast axis length
+! cl4_match_len specifies number of unique matches per observation
+! cl4_leadtime specifies the forecast axis lead time
+!
+&namcl4
+ cl4_match_len = 7
+ cl4_fcst_len = 3
+ cl4_fcst_idx = 1 2 3 1 2 3 1
+ cl4_vars = "forecast" "forecast" "forecast" "persistence" "persistence"
+ "persistence" "best_estimate"
+ cl4_leadtime = 12 36 60
+ cl4_prefix = "class4"
+ cl4_date = "20130101"
+ cl4_vn = "12.0"
+ cl4_sys = "FOAM"
+ cl4_cfg = "AMM7"
+ cl4_contact = "example@example.com"
+ cl4_inst = "UK Met Office"
+/
+\end{verbatim}
+\end{alltt}
+
+\subsubsection{Climatology interpolation}
+
+The climatological counterpart is generated at the start of the run by restarting
+the model from climatology through appropriate use of \textbf{namtsd}. To override
+the offline observation operator read routine and to take advantage of the restart
+settings, specify the first entry in \textbf{cl4\_vars} as "climatology". This will then
+pipe the restart from climatology into the output class 4 file. As in every other
+class 4 matchup the input file, input index and output index must be specified.
+These can be replaced with dummy data since they are not used but they must be
+present to cycle through the matchups correctly.
+
+\subsection{Advanced usage}
+
+In certain cases it may be desirable to include both multiple model fields per
+observation window with multiple match ups per observation. This can be achieved
+by specifying \textbf{nn\_ooo\_freq} as well as the class 4 settings. Care must
+be taken in generating the ooo\_files list such that the files are arranged into
+consecutive blocks of single match ups. For example, 2 forecast fields
+of 12 hourly data would result in 4 separate read operations but only 2 write
+operations, 1 per forecast.
+
+\begin{alltt}
+\tiny
+\begin{verbatim}
+ ooo_files = "F1.nc" "F1.nc" "F2.nc" "F2.nc"
+...
+ cl4_fcst_idx = 1 2
+\end{verbatim}
+\end{alltt}
+
+The above notation reveals the internal split between match up iterators and file
+iterators. This technique has not been used before so experimentation is needed
+before results can be trusted.
+
+
+
+
+\newpage
+
+\section{Observation Utilities}
+\label{OBS_obsutils}
+
+Some tools for viewing and processing of observation and feedback files are provided in the
+NEMO repository for convenience. These include OBSTOOLS which are a collection of Fortran
+programs which are helpful to deal with feedback files. They do such tasks as observation file
+conversion, printing of file contents, some basic statistical analysis of feedback files. The
+other tool is an IDL program called dataplot which uses a graphical interface to visualise
+observations and feedback files. OBSTOOLS and dataplot are described in more detail below.
+
+\subsection{Obstools}
+
+A series of Fortran utilities is provided with NEMO called OBSTOOLS. This are helpful in
+handling observation files and the feedback file output from the NEMO observation operator.
+The utilities are as follows
+
+\subsubsection{c4comb}
+
+The program c4comb combines multiple class 4 files produced by individual processors in an
+MPI run of NEMO offline obs\_oper into a single class 4 file. The program is called in the following way:
+
+\begin{alltt}
+\footnotesize
+\begin{verbatim}
+c4comb.exe outputfile inputfile1 inputfile2 ...
+\end{verbatim}
+\end{alltt}
+
+\subsubsection{corio2fb}
+
+The program corio2fb converts profile observation files from the Coriolis format to the
+standard feedback format. The program is called in the following way:
+
+\begin{alltt}
+\footnotesize
+\begin{verbatim}
+corio2fb.exe outputfile inputfile1 inputfile2 ...
+\end{verbatim}
+\end{alltt}
+
+\subsubsection{enact2fb}
+
+The program enact2fb converts profile observation files from the ENACT format to the standard
+feedback format. The program is called in the following way:
+
+\begin{alltt}
+\footnotesize
+\begin{verbatim}
+enact2fb.exe outputfile inputfile1 inputfile2 ...
+\end{verbatim}
+\end{alltt}
+
+\subsubsection{fbcomb}
+
+The program fbcomb combines multiple feedback files produced by individual processors in an
+MPI run of NEMO into a single feedback file. The program is called in the following way:
+
+\begin{alltt}
+\footnotesize
+\begin{verbatim}
+fbcomb.exe outputfile inputfile1 inputfile2 ...
+\end{verbatim}
+\end{alltt}
+
+\subsubsection{fbmatchup}
+
+The program fbmatchup will match observations from two feedback files. The program is called
+in the following way:
+
+\begin{alltt}
+\footnotesize
+\begin{verbatim}
+fbmatchup.exe outputfile inputfile1 varname1 inputfile2 varname2 ...
+\end{verbatim}
+\end{alltt}
+
+
+\subsubsection{fbprint}
+
+The program fbprint will print the contents of a feedback file or files to standard output.
+Selected information can be output using optional arguments. The program is called in the
+following way:
+
+\begin{alltt}
+\footnotesize
+\begin{verbatim}
+fbprint.exe [options] inputfile
+
+options:
+ -b shorter output
+ -q Select observations based on QC flags
+ -Q Select observations based on QC flags
+ -B Select observations based on QC flags
+ -u unsorted
+ -s ID select station ID
+ -t TYPE select observation type
+ -v NUM1-NUM2 select variable range to print by number
+ (default all)
+ -a NUM1-NUM2 select additional variable range to print by number
+ (default all)
+ -e NUM1-NUM2 select extra variable range to print by number
+ (default all)
+ -d output date range
+ -D print depths
+ -z use zipped files
+\end{verbatim}
+\end{alltt}
+
+\subsubsection{fbsel}
+
+The program fbsel will select or subsample observations. The program is called in the
+following way:
+
+\begin{alltt}
+\footnotesize
+\begin{verbatim}
+fbsel.exe