\documentclass[a4paper,10pt]{article}
%\usepackage{graphicx}
\usepackage{natbib}  % si appel à bibtex
%\usepackage[francais]{babel}
%\usepackage[latin1]{inputenc}  % accents directs (é...), avec babel
%\usepackage{rotating}

\setlength{\hoffset}{-1.in}
\setlength{\oddsidemargin}{3.cm}
\setlength{\textwidth}{15.cm}
\setlength{\marginparsep}{0.mm}
\setlength{\marginparwidth}{0.mm}

\setlength{\voffset}{-1.in}
\setlength{\topmargin}{0.mm}
\setlength{\headheight}{0.mm}
\setlength{\headsep}{30.mm}
\setlength{\textheight}{24.cm}
\setlength{\footskip}{1.cm}

\setlength{\parindent}{0.mm}
\setlength{\parskip}{1 em}
\newcommand{\ten}[1]{$\times 10^{#1}$~} 
\renewcommand{\baselinestretch}{1.}

\begin{document}
\pagestyle{plain}

\begin{center}
{\bf \LARGE 
Documentation for LMDZ, Planets version

\vspace{1cm}
\Large
The vertical discretization
}

\vspace{1cm}
S\'ebastien Lebonnois

\vspace{1cm}
Latest version: \today
\end{center}


\section{Theoretical aspects}

The position of the layers: 
\begin{itemize}
\item pressure limit between two layers, 
\item pressure within the layers
\end{itemize}

The Exner function: definition. 
It corresponds to the pressure levels within the layers.
Used for the computation of the potential temperature.
For the Earth, we use a specific scheme that computes these positions so that 
it maintains a condition of proportionality between total,
internal and potential energy (cf. a note from F. Hourdin). 

\section{Pratical aspects in the code}

\begin{itemize}
\item \textsf{disvert\_[no]terre.F[90]}: 
position of the interface pressure levels from an input file
(several possibilities). 
Definition of ap, bp and presnivs.
In the planetary version, definition of aps and bps.

This is done only once, called at the beginning from \textsf{iniconst.F}.

\item Interface pressures: 
computed in \textsf{caldyn0.F, caldyn.F, integrd.F, leapfrog.F}
through the \textsf{pression.F} routine.

\item Exner function (and therefore pressure within the layers): 
computed at three different places in \textsf{leapfrog.F} through the 
\textsf{exner\_[hyb/milieu].F} routine. 
For the Earth, we use \textsf{exner\_hyb.F}, that computes the positions in a
specific way to maintain a condition of proportionality between total,
internal and potential energy (cf. a note from F. Hourdin). 
For other planets, we use \textsf{exner\_milieu.F}, that computes the positions
of these pressure levels exactly in the middle of each layer. 
Though this fails to maintain the previous condition, there is no evidence of 
any significant influence on the results, and it makes it a lot easier to 
define correctly the level positions with the input file.
\end{itemize}

%\begin{thebibliography}{2}
%\providecommand{\natexlab}[1]{#1}
%\expandafter\ifx\csname urlstyle\endcsname\relax
%  \providecommand{\doi}[1]{doi:\discretionary{}{}{}#1}\else
%  \providecommand{\doi}{doi:\discretionary{}{}{}\begingroup
%  \urlstyle{rm}\Url}\fi

%\end{thebibliography}

\end{document}
