\subsection{Simulation of the radon cycle}

Radon is very appropriate to test
and compare the performances of transport and mixing in GCMs
\cite[see \eg][]{Genthon:1995,Prather:1993,Jacob:1990}.
$^{222}Rn$ is a chemically inert natural gas, and its only significant sink is 
radioactive decay with a period ($t_{1/2}$ = 3.8 days) typical of the
life time of tropospheric aerosols. $^{222}Rn$ is produced mostly in soils 
by the decay of ubiquitous $^{238}U$. It is exhaled at the surface of 
continents by exchange between the soil pores and the surface
air  \cite[]{Lambert:1982,Dorr:1984,Dorr:1990}.

A previous experiment with the LMD GCM gave broadly good results 
for the global $^{222}Rn$ distribution \cite[]{Genthon:1995}.
For this first work we used the Slopes scheme for large scale advection. 
The idea here consists in redoing  merely the same experiment but with 
Van Leer scheme I.

\subsubsection*{Models characteristics}

Three simulations are presented in this section.

The first one (Simulation A) is taken from \cite{Genthon:1995} and was
performed with the so-called "cycle5" version of the old
LMD GCM  \cite[see \eg][for the general performance
of this version]{Harzallah:1994}.
In this work, the Slopes scheme was used for radon advection 
and the parameterization of convection was that described previously
for the standard version of the LMD-Z model~:
the moist convective adjustment by \cite{Manabe:1965}
and a modified version of \cite{Kuo:1965} were applied sequentially.
Implementation of tracer transport for this simulation is described
in details in section 2.1 of \cite{Genthon:1995}.

The two other simulations
use the new LMD-Z model as described previously with Van Leer scheme I
(simulation A) and Godunov's scheme (simulation B) for advection of radon.
Because it is very difficult to represent properly the tracer flux with the 
so-called "Manabe-Kuo" convection scheme cited above, the choice
was made to introduce
the \cite{Tiedtke:1989} scheme in the tracer version of the LMD-Z model.

The Tiedtke scheme belongs to the category of "mass flux schemes".
 It  attempts to parameterize
explicitly the upward and downward fluxes in the convective tower
as well as the induced 
motions in the environmental air (Tiedtke, 1989).
Only one convective tower is considered where the entrainment and detrainment
between the cloud tower and the environment can take place at any 
level between the free convection level and the free sinking level.
The free sinking level is of course lowered by the entrainment and detrainment
processes. The convection closure is based on the large-scale moisture
convergence. 
The transport of tracer has been included in Tiedtke scheme following
that of water vapor.
With this last choice, we are able to predict explicitly the tracer
mixing ratio in updrafts and downdrafts in the convective tower.
A more complete description of the tracer parameterization in
this convective scheme will be given in a forthcoming paper by
Armengaud et al..

The main characteristics of the three simulations are summarized in Table
\ref{tab:sim}.

From \cite{Genthon:1995},
we keep the results of a simulation which uses the Slopes scheme
and the soil temperature control. 
It means that the soil-atmosphere exchange of radon is prevented where and
when the soil temperature is below 0~\(^{\circ} \)C
\cite[Simulation LMD2 in][]{Genthon:1995}.

For the three simulations, the horizontal grid is the standard constant
area grid of LMD ``cycle 5'' with 64 points regularly spaced in longitude
and 50 points regularly spaced in sine of latitude.

\vspace{0.4cm}

\begin{table}[htbp]
\begin{center}
\begin{tabular}{cccc} \hline 
\\
\em{Case} & \em{GCM} & \em{Advection} & \em{Convection}  \\
\\
\hline
\\
A  &  LMD"cycle5" & Slopes        & Manabe-Kuo \\
B  &  LMDZ        & Vanleer I     & Tiedtke    \\
C  &  LMDZ        & Godunov       & Tiedtke    \\
\\ 
\hline
\end{tabular}
\end{center}
\caption{Summary of the simulations}
\label{tab:sim}
\end{table}

The tracer models for each case A, B and C  are integrated over 27 months.
To account for model spin-up, the first three months are ignored in
the results presented below.


\subsubsection*{Results}

\fg{radon} shows the global distribution of $^{222}Rn$ in the first 
layer for the three cases. A very strong similarity between cases 
A and B is observed.   
Above continents the weak difference observed in South America
or in Africa can be attributed
to the use of different convections schemes.
Above the oceans the shapes of the $^{222}Rn$ isolines are pretty
much the same in experiments A and B.
Sharp gradients in the surface concentrations of $^{222}Rn$ are usually
observed close to the coasts (Lambert et al., 1982, henceforth L82). 
These sharp gradients, already well represented with the slopes scheme (A),
are equally well reproduced with the Van Leer I scheme (B).
The discrepancy between these two cases
observed over North Atlantic Ocean and Arctic Ocean are not significant due
to the very coarse latitudinal resolution with a constant-area grid in this
region.

Case C shows much weaker gradients in South Atlantic and off the
North-West American coast. The diffusive effect of the Godunov scheme is indeed 
obvious on the lower panel of \fg{radon}. Isolines 2.5 and 5 pCi/m$^{3}$
are much further away from the coasts than in the cases A and B.  

\begin{table}[htbp]
\begin{center}
\begin{tabular}{ccc} \hline 
\\
\em{Source} & \em{Region A} & \em{Region B} \\
\\
\hline
\\
L82 (observations)     &   5      &  1        \\
                       & (3-10)   & (0.5-1.8) \\
CASE A  (Slopes)       &   4.6    &  2.7      \\
                       & (0.2-31) & (0.2-34)  \\
CASE B  (VanLeer)      &   5.2    &  2.9      \\
                       & (0.4-35) & (0.3-25)  \\
CASE A  (Godunov)      &   9.2    &  4.5      \\
                       & (0.6-65) & (0.5-40)  \\
\\ 
\hline
\end{tabular}
\end{center}
\caption{Simulated mean $^{222}Rn$ Surface Concentrations compared with
observations by Lambert et al. (1982, L82).
Numbers in parentheses indicate the spatial variability about the mean.
Region A: Atlantic and Indian Oceans north of $20^{\circ}$N plus Pacific Ocean 
north of equator. Region B: other oceanic regions.
\label{tab:res}}
\end{table}

\subsubsection*{Simulations versus observations}

Cases A and B are in good agreement with the Larson's and Lambert's
measurements over the Atlantic and Pacific Oceans in general.  $^{222}Rn$
concentrations off the West 
Coast of the United States are lower than off the East Coast. 
The more southerly the crossing 
of the North Atlantic, the lower the radon 
 \cite[]{Larson:1978,Larson:1979,Larson:1980,Hoppel:1990}.
The same 3 to 5 pCi/m$^{3}$ of radon is
also found in maritime air fairly close to the West coast
of North America \cite[]{Larson:1979,Larson:1980,Hoppel:1984,Hoppel:1985,Hoppel:1990}.
In general, discrepancies with observed values are more important in
case C for these typical
oceanic regions. This can be confirmed with the analysis of
Table~\ref{tab:res} which
presents averaged $^{222}Rn$ surface
concentrations for two oceanic regions, according to L82~:
region A is defined by  Atlantic and Indian Oceans north of 20~\(^{\circ} \)N
plus Pacific Ocean 
north of equator; region B is defined by the other oceanic regions.
Case A and B are very close to each other, both for the mean and the
variability whereas case C gives values about twice as large, due to a larger 
numerical diffusion.

Over region A, the mean values for case A and B are in very good agreement
with L82
whereas, over region B, even
case A and B overestimate the surface radon concentrations.
This discrepancy can be related to the large
difference between our simulations and data sets
in the South Atlantic Ocean off
Argentina where observations give 1 pCi/m$^{3}$ or less 
against 2 to 20 pCi/m$^{3}$ for the simulations.
A Larson's hypothesis would attribute this low
radon over South Atlantic
to very low radon concentrations in the surface soil of Argentina
\cite[]{Larson:1980}.

A complete description of the representation of the radon cycle in the
new LMD-Z model will be given in a forthcoming publication.

\begin{figure}
\vspace{-1.5cm}
\centerline{\includegraphics[width=14.cm]{\EPS/alex.eps}}
\caption{2-year means of the distribution of $^{222}Rn$ in
the first model layer with
the LMD-cycle~5 GCM (A : Slopes scheme).
and
the LMDZ GCM (B : Van Leer I and C : Godunov).
Soil control temperature and same resolution in all cases.
\label{fg:radon}}
\end{figure}