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\title{Numerical Modelling of the Circulation of Superrotating Atmospheres: Venus and Titan 
\author{F. Hourdin$^1$, O. Talagrand$^1$, K. Menou$^1$,
       R. Fournier$^1$, J.-L. Dufresnes$^1$, D. Gautier$^2$,\\
 R.Courtin$^2$, B. Bézard$^2$ and C.P. McKay$^3$\\ 
 $^1$Laboratoire de M\'et\'eorologie Dynamique du CNRS, Ecole Normale Sup\'erieure,\\
 24 rue Lhomond, 75 231, Paris Cedex 05, France (hourdin@lmd.ens.fr)\\
$^2$ DESPA, Obs. Paris-Meudon, Meudon, France\\
$^3$ NASA/Ames Research Center, Palo-Alto, California, USA.
 \date{\today}
 }}

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\begin{abstract}
Atmospheric superrotation has been known for decades on Venus where the cloud cover, at 70~km above the surface, rotates about 60 times faster than the solid planet. Superrotation is also strongly suspected on Titan on the basis of both numerical studies and observations. We show how general circulation models, originally developed for terrestrial meteorology and climatology, can be used to infer the circulation in such atmospheres. We focus on the case of Titan for which a specific GCM has been developed at LMD in the last few years. 
\end{abstract}

\section{Introduction}


As explained in the accompanying paper by Hourdin et al. ``Numerical modelling of the general circulation of the Martian atmosphere'', General Circulation Models (GCM's), originally developed for 
the needs of terrestrial meteorology and climatology, have been intensively used on Mars.
Numerical results obtained with Martian GCM's are generally in very good agreement with available observations and those models have played a central role in the study of the Martian climate in the past 20 years. We have shown also in the accompanying paper how they can be used for the preparation of spacecraft exploration of Mars.

In fact, GCM's developed for the terrestrial atmosphere can be adapted without major difficulties
to the conditions of the other terrestrial planets.
However, this approach has been much less productive in the case of Venus.
The main intriguing feature of the Venusian atmospheric circulation is known as atmospheric superrotation :
at the cloud level, about 70~km above the surface, the atmosphere as a whole rotates about 60 times faster than the solid planet with a rotation period of about 4 terrestrial days.

Understanding of superrotation is still an open question.
Past models of Venus did not produce superrotation except with ad hoc \cite[]{Youn:77} or non physical sources of angular momentum (see e.g., Tourte, 1984). \nocite{Tour:84}
In the last few years however, two teams have obtained superrotation in GCM's
\cite[]{Hour:92b,Del:93} in idealized conditions.
Of course, the fact that GCM's are able to simulate superrotation helps to understand this phenomenon since many diagnostics can be performed in the models which cannot be performed from observation, in particular the decomposition of the atmospheric angular momentum budget in terms of mean meridional circulation and eddies.

In the frame of the join NASA/ESA mission Cassini-Huygens to the Saturn system, LMD has developed a GCM for Titan.
Simulations performed with the Titan atmospheric GCM also produced a strong stratospheric superrotation
with prograde winds of the order of 100 m/s at the equator and an upper stratosphere rotating about 10 times faster than the solid planet (the rotation period of Titan, assumed to be locked with Saturn, is 16 days). Those results are in good agreement with some indirect observations.

The paper presents results both from the idealized simulations of superrotation and for the case of Titan and ends by some concluding remarks including the need to come back to the simulation of the Venusian atmospheric circulation.



\section{Numerical simulation of atmospheric superrotation}

In the last few years two teams have obtained atmospheric superrotation in GCM's for idealized conditions. In the case of LMD the results were
obtained with a simplified GCM developed for that purpose
\cite[]{Hour:92b}. Similar results were obtained at NASA/GISS with a dry version of a terrestrial GCM with a cloud deck at the top of the model.



As explained in the accompanying paper by Hourdin et al.
 (``Numerical modelling of the general circulation of the Martian atmosphere''),
 GCM's are based on the numerical integration of the equations of
hydrodynamics on the sphere. This part of the code is general enough to be applied to all
atmospheres of terrestrial type (i. e. atmospheres whose depth is small in comparison to the
planetary radius, and which remain vertically in hydrostatic balance on large horizontal
scales). GCM's also contain a set of "physical parametrisations": mainly a representation of
small scale turbulent mixing in the Planetary Boundary Layer and a representation of
absorption, emission and scattering of radiation (solar and thermal infrared). Contrary to the
purely 'dynamical' part of GCM's, the representation of radiation is very specific to each
planet, and may require substantial work when adapting a model to the conditions of a new
planet. 

We have developed at LMD, in addition to the original terrestrial model and to the Martian version,
a more general version of the GCM
in which the entire description of the planet is reduced to a set of 19 parameters.
The main step in developing this model has been the choice of a simple parametrisation for radiative transfer : for both solar and thermal infrared radiation, we use a kind of grey approximation so the radiative code only depends upon 2 independent parameters~: the total atmospheric transmission for each part of the spectrum.


\input{tbparam}


The list of the 19 independent parameters, together
 with typical terrestrial values is given in \tb{param}.

\begin{figure*}
\begin{center}
\includegraphics[width=13cm]{post/uterre.epsi}\\
\includegraphics[width=13cm]{post/usup.epsi}
\end{center}
\caption{Zonally averaged zonal wind as simulated with the 19-parameter GCM. Upper panel : terrestrial values of the parameters given in Table~1 (average over December-January-February). Lower panel : same parameters but with a zero obliquity, a ten times smaller planetary rotation rate and a larger absorption of solar radiation by the atmosphere ($\tau_{Vis}=0.1$).\label{fg:u}}
\end{figure*}

When run with the values of parameters given in the table, the model produces results in relatively good agreement with the Earth climatology. In particular, the zonal wind (upper panel of \fig{u}) shows a structure with two jets in mid-latitudes, with about the right intensity (the winter jet being stronger than its summer counterpart), and easterlies in the tropics. The upper atmosphere circulation is very badly represented because of the absence of a  tropopause. Note that oceans are just represented by using a very large thermal inertia for the surface\footnote{In the model, the surface temperature is computed from the surface fluxes by integrating the equation of thermal conduction in the soil.}
(see \tb{param}).

By changing only three parameters, the model produced a strong superrotation as illustrated in the lower panel of \fig{u}.
For this simulation, the obliquity was set to zero (which simplifies the analysis because of the absence of seasons and makes the planet somewhat similar to Venus), the rotation of the planet was divided by ten and the absorption of solar radiation by the atmosphere was significantly increased. This last effect tends to stabilize the vertical thermal profile, which in turn decreases the vertical damping of superrotation by turbulent mixing.

\begin{figure}

\includegraphics[height=8cm]{post/gierasch.epsi}

\caption{Schematic mean meridional circulation (heavy line) in the simulation of superrotation and mean meridional transport of angular momentum (white arrows).\label{fg:mmc}}
\end{figure}

The mechanism responsible for superrotation in the model is that originally proposed by \cite{Gier:75} for Venus.
In the model, the mean meridional circulation is dominated by a large equator-to-pole Hadley cell as illustrated in \fig{mmc} whereas the Hadley circulation is confined to the tropical regions on Earth. This difference is essentially due to the slower rotation rate of the planet.
Gierasch noticed that, as soon as  angular momentum (product of the east-west component of the absolute wind by the distance to the polar axis) decreases from the equator toward the pole (for instance in an atmosphere at rest with
respect to the rotating planet), a
Hadley cell transports more angular momentum upward near the equator than downward near the poles. This is illustrated in \fig{mmc}.

Gierasch also noticed that, in a superrotating atmosphere, such a Hadley cell tends to transport systematically angular momentum from equatorial to polar regions since there is more angular momentum transported poleward in the upper branch of the Hadley cell than equatorward near the surface (where the zonal wind is expected to be smaller because of surface friction). Thus, in order to complete the explanation for superrotation, one must find a process able to transport angular momentum equatorward. In the simulation, this transport is clearly done by large scale quasi-barotropic eddies. Moreover, those eddies are shown to be correlated with regions in which the necessary condition for barotropic instability (change of sign of the latitudinal derivative of potential vorticity) is satisfied (not shown).
In fact, there is a simpler reason why such a mechanism must occur in some way~: the fact that angular momentum increases poleward somewhere in the flow is a sufficient and necessary condition for inertial instability. As a consequence, if somewhere in the flow angular momentum exceeds that of the air situated closer to the equator, angular momentum will rapidly mix restoring the maximum at the equator.

\section{Titan}

\begin{figure}
\includegraphics[height=8cm]{post/occult.eps}
\caption{Latitudinal zonal wind profile deduced by Hubbard et al. (1993)
from the 28-Sgr occultation which corresponds to a pressure level near
0.25~mbar (squares). The other three curves show the zonally averaged
zonal wind as produced by the LMD Titan GCM for the same season (northern
summer)
and for three different pressure ranges (Hourdin et al., 1995).
\label{fg:utitan}}
\end{figure}

We also developed at LMD the first GCM of the atmosphere of Titan by introducing the radiative transfer code developed for 1-dimensional studies by \cite{McKa:89,McKa:91}.
When integrated over very long periods (25 Titan years or about 750 terrestrial years) the Titan GCM also predicted
strong prograde zonal winds near the 1-mbar level, of the order of 100 m/s \cite[]{Hour:95b}.

This results are very consistent with the only indirect estimates we have of zonal winds in the stratosphere of Titan. The first estimate comes from the thermal wind balance computed from the temperature retrieved from the voyager IRIS observations. This analysis suggested wind of the order of 100 m/s in the 1-mbar pressure range \cite[]{Flas:81}. More recently, a zonal wind
profile with similar velocities
was derived for the same pressure range from ground-based observations of the occultation of a
very bright star by Titan \cite[]{Hubb:93}. The comparison of this profile to model results is shown in \fig{utitan}.
The agreement is very good for the equatorial wind but the model seems to underestimate the strength of the mid-latitude winter jet (note however that the oscillations in the observations are an artefact of the retrieval method).

As in the idealised simulations described above, the
superrotation is created in the Titan simulation by a combination of upward transport of
angular momentum by the mean meridional Hadley circulation and
equatorward transport in the stratosphere by large scale eddies.
Once again, the model shows that the waves responsible for the
equatorward transport of angular momentum are created by dynamical
unstabilities on the equatorward flank of the high latitude jets.

However, the model also predicts latitudinal temperature gradients much
weaker than observed (not shown). This deficiency is attributed to a lack of
latitudinal forcing in the model which does not take into account
the latitudinal variations of the chemical composition and aerosol
content.
 
There has been a large debate on whether the hemispheric asymmetry of
temperature observed by Voyager was due to these latitudinal
variations of composition (e.g. B\'ezard et al., 1995) \nocite{Beza:95}
or to dynamical effects \cite[]{Flas:90}.
 The basis of this debate was that the temperature should
have been symmetric with respect to the equator at the time of the Voyager
encounter (equinox) since the radiative time constant is much shorter
than one year.
In fact, recent radiative computations with a full seasonal cycle but
without dynamics and without any latitudinal variation of composition
have produced a small hemispheric temperature contrast in the
stratosphere which is in surprisingly good agreement with Voyager observations \cite[]{Hour:95b}.
Although these results could theoretically close the debate (nothing else is
needed to explain the latitudinal contrasts) it appears more and more
clearly that the understanding of the physics of the upper atmosphere
of Titan involves coupling processes between dynamics, aerosol physics and
chemistry. It has already been shown by \cite{Hutz:95}
that dynamics could explain the magnitude of
the hemispheric albedo contrast observed by Voyager.


\section{Back to Venus...}

Following the preliminary work by \cite{Tour:84}, we now plan to develop a full
GCM of the atmosphere of Venus. The development of a specific radiative code is already
under way. The first scientific goal is to validate the simulations of atmospheric superrotation
against the observations, which are relatively much more abundant on Venus than on Titan.
Of course, a GCM of the Venusian atmosphere could also be used in the future for the
preparation and exploitation of spacecraft missions.

\section{Concluding remarks}

The GCM of Titan, the development of which has been motivated by the preparation of the Huygens mission, is probably still rather far from completely and accurately representing the complex and mysterious climate of Titan. Despite this probable weakness, the model results have already been used for the preparation of the Huygens mission, for instance as a basic state for computing the intensity of possible gravity waves. This is due of course to the lack of other sources of information, the best observation we have on winds being the very indirect retrieval shown in \fig{utitan}.

We are now engaged in the development of a much more complete model including interactions between atmospheric dynamics, aerosol physics and chemistry. 
The full prediction of this complex world and the confrontation
to remote and in-situ measurements at the time of the Cassini-Huygens
mission is probably one of the most challenging subject for the
coming years in planetary sciences.

\bibliography{/d2/hourdin/tex/biblio/fred}

\end{document}

