
\def\FIGURES{\local/FIGURES}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Introduction}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

The atmosphere of Titan, which will be soon observed by the Cassini/Huygens
mission, is one of the most mysterious and exciting climate systems of
the solar system.
Titan's dense nitrogen atmosphere (1.4 atm at the surface for a gravity
of 1.35~m~s$^{-2}$) is thought to be dominated by a strong zonal flow,
peaking in the
stratosphere at around 200-300~km. At these altitudes, the atmosphere is
rotating about 10 times faster than the solid body, itself locked in phase 
around Saturn, with a 16-day period. Composed mainly of nitrogen (90-98\%)
and methane (2\% above tropopause), 
Titan's atmosphere is also the site of an intense photochemistry, initiated
between 600 and 1000 km by dissociation of  molecules
by magnetospheric electrons and UV photons. Laboratory experiments suggest
that compounds as complex as ``ammono" analogs to amino acids
could be present \cite[]{Raul:02}.
About 20 different species have been detected thus far, either by Voyager
or from ground based observations.
This rich photochemistry yields polymerization of hydrocarbons and nitriles,
with formation of a dense aerosol layer in the stratosphere, that gives Titan
its orange color on Voyager visible images. The details of this
polymerization are not yet understood, and both experimental and theoretical
studies are in progress on this point 
\cite[]{Coll:99,Tran:03I,Lebo:02,Wils:03}.

Almost all the gaseous species show a strong enrichment in
the northern polar latitudes at the time of Voyager encounter,
shortly after northern spring equinox.
If defined as the ratio between the average concentration at 50-70 N
and that at 0-30 N, this enrichment varies from a factor of 1.4
to more than 20 depending on the particular chemical compound.

In a review of the composition of Titan's atmosphere from a meteorological
viewpoint, \cite{Flas:98} suggests that the meridional transport and
subsidence of stratospheric species could play a significant role in
enhancing the polar concentration. From meridional winds of 4~m~s$^{-1}$
and a dynamical time scale $\sim 4$~terrestrial years, he estimated an HCN
enhancement of a factor of 2, and concluded that some additional factor,
such as a localized meridional circulation, was necessary to explain the
observations. He also discusses the possibility of the circumpolar vortex
becoming dynamically and compositionally isolated from the remainder of
the atmosphere during late winter and early spring. In a speculative
analogy with Earth's ozone hole, the isolation would end with the breaking
of the vortex by planetary waves, which would mix constituents, heat and
vorticity across the boundary.

Combining a photochemical model and transport of atmospheric compounds by 
meridional circulation, \cite{Lebo:01} have clearly attributed the
enrichment of northern high latitudes at the time of Voyager to the
downward advection of chemical compounds from the production zone,
in the downward branch of a global pole-to-pole Hadley cell which prevails
almost until northern spring equinox.
Without additional factors, the coupled model was able to
reproduce even the large value observed for the 
polar enrichment of HCN.
No clear explanation was given however for the wide range observed
for the magnitude of this polar enrichment.
\cite{Lebo:01} suggested that this magnitude could be partly controlled
by chemistry: a stronger
contrast on the vertical, due for instance to a stronger source in
the upper atmosphere relatively to the sinks, could produce, through
downward advection in high latitudes, a stronger
polar enrichment.



The present paper casts a new light on this question.
For most species, the magnitude of the latitudinal contrast
does not depend in fact on the actual chemistry but
only on the altitude at which they condense.
We show that the very same latitudinal contrasts can be obtained
either with a full chemical model forced by the GCM dynamics or with
a very simple tracer fixed to one in the upper stratosphere and undergoing
advection and condensation at a given level.
It is shown also that advection in
the meridional plane and latitudinal mixing by waves is controlling
the vertical gradient of chemical compounds.

If atmospheric transport is responsible for the latitudinal and vertical
variations of the stratospheric composition,
then observations of composition can also provide valuable information
on the atmospheric circulation.
In particular, the good agreement of model results and observations in terms
of  polar enrichment and vertical profiles
provides a strong validation of the atmospheric dynamics given that
this polar enrichment is quite sensitive to the
model dynamics. In fact, polar enrichment in itself is evidence for the
existence of the Hadley circulation on Titan. This fundamental component of
the dynamics has never been directly observed on Titan.
Observations of the latitudinal and time variations of atmospheric
composition by the Cassini orbiter will thus contain very important
constraints on the atmospheric circulation.

The paper begins with a synthesis of previous modelling work on
the simulation of the atmospheric composition and dynamics of Titan's
atmosphere including a description of the version of the coupled
model used in this study (section 2). We then discuss what
controls the vertical and latitudinal variations of atmospheric
chemical compounds (section 3) and how their simulation depends
on the accuracy of the simulated atmospheric dynamics (section 4).
Seasonal variations are discussed in section 5.
We finally present some predictions of what Cassini should
observe during the mission and underline the importance an extended
mission covering a longer fraction of a Titan year could have
(section 6) before some concluding remarks.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Titan Climate Modelling}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\subsection{Atmospheric Superrotation}

\cite{Goli:75} was the first to suggest that, like Venus, Titan
may be the site of an intense atmospheric superrotation. 
Observations of latitudinal temperature contrasts by the Voyager~1 probe
\cite[]{Flas:81}  as well as the shape of Titan's atmosphere 
%[reconstructed from ground based observations of the occultation of a 
%very bright star, {\em Hubbard et al.}, 1993]
\cite[reconstructed from ground based observations of the occultation of a
very bright star, ][]{Hubb:93}
both suggested a strong zonal flow, but with no constraint on its direction.

The prograde nature of winds was first predicted theoretically and simulated
numerically with a general circulation model
adapted from the climate model of the Laboratoire de M\'et\'eorologie
Dynamique \cite[]{Hour:92b,Hour:95b}, LMDZ.
It has since gained mounting observational
confirmation \cite[]{kostiuk2001, moreno2003, bouchez2003, luzdps2003}.

The adaptation of the climate model from Earth to Titan mainly consisted in
turning off the water cycle and replacing the radiative transfer computation by
the code developed for Titan by \cite{Toon:89} for the solar
range (UV and IR) and \cite{McKa:89} for the thermal IR.
In this first study \cite[]{Hour:95b}, the radiative effect of latitudinal variations
of the atmospheric composition was ignored.
The 3-dimensional dynamical core, based on the primitive equations of meteorology, 
was kept unchanged except for parameters such as
gravity or the thermodynamic gas constant.
Those simulations have been preceded by simpler studies 
\cite[]{Hour:92b,Del:93,Hour:96estec} for idealized
planetary conditions in which superrotation has been obtained for
planets rotating much slower than the Earth, 
at zero obliquity (as on Venus) and increased absorption of solar radiation
in the visible.

Both for the realistic Titan case and for the simpler simulations mentioned
above, the simulated superrotation
is explained by the so-called Gierasch-Rossow mechanism \cite[]{Gier:75}.
The net upward transport of angular momentum (product of the absolute
zonal velocity and 
the distance to the polar axis) by global equator-to-pole Hadley cells
is able to create and maintain an excess
of angular momentum in the upper atmospheric layers, if the angular
momentum is larger in low than in high latitudes (this is the case for
instance for an atmosphere at rest with respect to the surface).
Once superrotation sets in, there is also more angular momentum
transported into high latitudes in the upper branches of the Hadley cells
than equatorward near the surface.
This net poleward transport of angular momentum is balanced by equatorward
transport by non-axisymmetric eddies as originally suggested by
\cite{Gier:75} and \cite{Ross:79} in the context of Venus. In the numerical simulations,
those eddies were identified as planetary waves forced by barotropic and
inertial instabilities on the equatorward flank of the strong high latitude
jets created, like the jet stream on Earth, by the poleward transport of
angular momentum from the equator.
In this context, the slow planetary rotation rate is important
to allow the Hadley circulation to extend up to the pole \cite[]{Hunt:79,Del:87}
while the increased static stability reduces the coupling between the surface
and the superrotating atmosphere.

The main difference between Titan simulations \cite[]{Hour:95b} and
the above mentioned simulations of atmospheric superrotation is the
strong seasonal cycle arising from Titan's large obliquity
(26.7$^\circ$): during almost half a year around solstice, the
meridional circulation is dominated by a Hadley cell extending between
both poles, with rising motion in the summer hemisphere. This
meridional circulation creates an intense circumpolar jet in the
winter hemisphere, analogous to the jet stream on Earth.  It is only
close to the equinoxes or on an annually averaged view that the
meridional circulation is dominated by two equator-to-pole Hadley
cells.

For more details see the introduction of \cite{Luz:03I}.

\subsection{Coupling with Atmospheric Composition}


Those first simulations of Titan's atmospheric circulation
strongly underestimated the latitudinal
temperature contrasts measured by Voyager in the stratosphere
as well as the intensity of the circumpolar jet inferred from
the stellar occultation. \cite{Hour:95b} suggested
that this deficiency could be due to the
fact that the radiative computations did not take into account the latitudinal
contrasts in atmospheric composition
or haze opacity. 

Modelling work and analysis of Voyager observations did
confirm the strong coupling between chemistry, microphysics and stratospheric
dynamics.
By forcing a microphysical model of Titan's haze by the meridional circulation
predicted with the Titan's GCM, it was shown that meridional transport
alone could explain the order of magnitude of the observed hemispheric and
seasonal  contrasts in Titan's albedo \cite[]{Hutz:95}.
With a similar approach, the polar enrichment in gaseous compounds at the
time  of the Voyager encounter was attributed to downward advection from
source regions in the subsiding branch of the Hadley cell \cite[]{Lebo:01}.
Finally, from radiative computations, it was shown that the latitudinal 
variations of atmospheric composition and haze opacity, 
as deduced from Voyager observations,
were able to modulate significantly the radiative balance of Titan's
stratosphere \cite[]{Beza:95}.

\subsection{The 2-Dimensional Coupled Model}

\begin{figure*}
\centerline{
\includegraphics[angle=-90,width=5.5cm]{\FIGURES/profils_a.eps}
\includegraphics[angle=-90,width=5.5cm]{\FIGURES/profils_b.eps}
\includegraphics[angle=-90,width=5.5cm]{\FIGURES/profils_c.eps}
}
\caption{Comparison between simulated and observed stratospheric composition
at solar longitude L$_s$=9 (Voyager 1 fly-by).
Model results including full chemistry is plotted with solid lines. Dashed lines are
for the linearized chemistry (see text). Black circles are observations from
Voyager 1 (Coustenis and B\'ezard, 1995). Pressure levels correspond to the
maximum of the contribution function (at equator) for each compound.
For the model, concentrations are extracted at this particular pressure level.
Observations for HC$_3$N at low to mid latitudes are upper limits.
\label{fg:compo}}
\end{figure*}

For the reasons above, it was then decided to develop a climate model
for Titan, coupling dynamics, chemistry and haze microphysics.
Because of the prohibitive numerical cost of the 3-dimensional Titan GCM,
the model was developed based on a restriction to axi-symmetric component.
The microphysical model by \cite{Caba:92} was introduced
in the climate model based on a description of the haze particles in terms
of 10 bins in radius (instead of the 45 bins of the original 1-dimensional model).
%\cite[instead of 45 bins in the original 1-dimensional model, ][]{Caba:92}.
The smallest size is for the freshly created macromolecules ($r_1=1.64\times 10^{-9}\mbox{ m}$), the other radii being given by $r_{n+1}=16^{1/3}r_n$.
The monomer radius is $r_m=66\mbox{ nm}$ and the volume ratio between two adjacent bins is 16. Particles larger than $r_m$ are represented as fractal aggregates.
%The first five  bins correspond to spherical particles.
%The fifth one serves as a monomer for aggregates particles represented
%as fractals (5 last bins).
The radiative code was upgraded to account for the coupling with
the new aerosol description \cite[]{Rann:04}.
The atmospheric composition is handled by a module consisting essentially of the photochemical
model of \cite{Lebo:01}, which has been integrated in the GCM. This yields
a stratospheric composition in rather good agreement with available
observations, as shown in Fig.~\ref{fg:compo}. 
A detailed analysis of the most significant discrepancies
(C$_2$H$_6$, C$_3$H$_8$, C$_4$H$_2$, HC$_3$N), and comparison with the
previous results of \cite{Lebo:01} is not in the scope of this paper,
and will be published in a forthcoming study.
As in \cite{Lebo:01}, linear tracers can also be used for
C$_2$H$_6$, C$_2$H$_2$ and HCN (thereafter named linearized chemistry).
The chemical production and loss terms are replaced by relaxation
toward a reference profile with a given photochemical time constant, both taken
from a 1-dimensional photochemical model.
The advantages of this simplified chemistry are twofold: 1) the code
is faster and 2) the model can be tuned so as to reproduce the
observed mean stratospheric concentrations. This is important
when evaluating  the effect of chemical composition on atmospheric
dynamics.
Latitudinal profiles
obtained with the linearized chemistry for these three compounds
are also shown in Fig.~\ref{fg:compo}.
In the present study, both the full and linearized chemistry are used.

When using the full chemistry, the upper boundary has to be chosen for all
species. Chemistry taking place above the GCM limits has a crucial role
for most constituents, which are produced in the mesosphere and then
transported downward. For the simulations used in this paper, we decided 
to fix the exchange flux between the upper atmosphere and the upper layer 
of the GCM. These fluxes are calculated using a 1-dimensional model of 
Titan's atmosphere in equatorial conditions \cite[see ][]{Lebo:02,Lebo:03a}.
As a first approximation, these fluxes are fixed, both in time and as a
function of latitude. A second option is to fix the composition in the upper
layer of the GCM. This option was also tested, again with no temporal or
latitudinal variations, and it gives results very similar to our first option.
For the linearized chemistry, the upper boundary condition is handled 
in a different way: the concentration is forced to stay close to the
1-dimensional reference profile in the three upper layers
by imposing arbitrarily short chemical time constants
(1 Titan day in the top layer, 10 days in the second and 100 days in the
third layer from the top).
The upper condition essentially controls the mean stratospheric concentration
without affecting latitudinal contrasts, as will be shown below.

To avoid difficulties related to the possible impact of temperature
errors, condensation is computed for each species
(both for the full and linearized
chemistry) from a concentration at saturation
estimated once at the beginning of the simulation
using the temperature profile of \cite{Lell:89}.

One important point when reducing the model to a 2-dimensional latitude-altitude
version, is to parametrize the effect of latitudinal transport of
momentum, heat, or tracers by planetary waves. For this, a specific
parametrization was developed \cite[]{Luz:03II}. In this parametrization,
latitudinal mixing by waves is prescribed as an eddy diffusion.
The eddy diffusivity $K_y$ itself is parametrized as a function of the
explicit variables of the axi-symmetric model (closure).
This parametrization was developed and tuned based on a series of
simulations of planetary waves in Titan-like conditions
with a global shallow water model
\cite[]{Luz:03I}. The diffusivity $K_y$ mainly depends
on the barotropic instability of the mean zonal flow.

For the simulations presented here, the model discretization
is based on 48 latitude bands of equal width spread from pole to pole with 55
vertical layers of which the last three serve as a sponge layer to
dump down the wind and prevent spurious reflection of waves. Level 52 is
at approximately 480~km above the surface.
The vertical resolution is about 3~km in the troposphere, 5~km
at the tropopause and 10-15~km in the stratosphere, which corresponds to
one half to one third of a scale height.
The dynamical equations are integrated with a time step of 3 minutes.
The radiative transfer is computed ten times per Titan day.
Additional informations can be found in \cite{Luz:03II} and \cite{Rann:04}.

All the simulations presented here were started from an initial state
inherited from previous pluriannual simulations.
The model was integrated for several Titan years and it was checked at the end,
by comparing two successive years, that the results presented were identical.
It appears in fact that the time constants involved when studying the 
stratospheric dynamics and composition are relatively short.
In particular, the time constants for composition are not determined by
the reaction rates themselves, which can correspond to very long time
constants, but rather by transport and condensation close to the tropopause
as analyzed in detail below. Finally, the results presented below for
sensitivity experiments correspond
to year 6, which is more than enough to obtain stable results from one
year to the other.



\subsection{Impact of Coupling on the Dynamics}

\begin{figure}
\centerline{
\includegraphics[width=8cm]{\FIGURES/tcoust.eps}
\includegraphics[width=8cm]{\FIGURES/uhubb.eps}
}
\caption{Stratospheric temperature (at 1~hPa, L$_s$=9) and zonal winds
(at 0.25~hPa, L$_s$=128).
Comparison between observations and simulations with and
without coupling with the atmospheric chemistry and haze microphysics.
Temperatures were retrieved from infrared remote sounding by Voyager
shortly after northern spring equinox.  The observed wind profile
was deduced from the 1989 stellar occultation.  The symmetric shape is imposed
by the retrieval procedure but observations are only reliable in
the shaded area.
\label{fg:Voyager}}
\end{figure}

The first important result of the coupled model was to explain the 
origin of the detached haze layer observed in Voyager images above the
main haze layer and to predict a strong accumulation of haze in the
polar night \cite[]{Rann:02}. This accumulation in turn increases radiative
cooling to space in high latitudes,
enforcing the meridional forcing of the general circulation \cite[]{Rann:04}
and hence latitudinal temperature contrasts and the zonal circumpolar jet,
in better agreement with observations.
The coupling with chemistry has a similar effect but occurs higher
in the atmosphere, and is less sensitive for comparison with available
observations \cite[]{Lebo:03}.

As an illustration, we show in Fig.~\ref{fg:Voyager}
the comparison between the simulated and
observed zonal wind profiles 
%[comparison with the occultation
%profiles close to northern summer solstice, {\em Hubbard et al.}, 1993] 
\cite[comparison with the occultation
profiles close to northern summer solstice, ][]{Hubb:93}
and latitudinal temperature profiles at 1~hPa \cite[compared with Voyager
retrieval close to northern spring equinox by][]{Flas:81,Flas:90,Cous:95}.
The comparison is shown for two simulations, one without coupling and another with the dynamics coupled
with both haze and chemistry.
The temperature contrasts in winter
mid latitudes and the jet speed are in much better agreement for the coupled case.

\subsection{North/South Temperature Asymmetry}


Note also that the coupled simulation tends to produce a north/south
asymmetry of temperatures (upper panel of Fig.~\ref{fg:Voyager}) consistent with Voyager observations.
Voyager observed, shortly after northern spring equinox,
a warmer southern hemisphere.
This observation was a surprise since radiative
time constants were estimated to be much shorter than a season
at the pressure levels sounded by the methane bands. 
It was first suggested that this asymmetry could be due to a phase lag in
the latitudinal redistribution of angular momentum by the mean meridional
circulation \cite[]{Flas:90}.
Full simulations with a radiative convective model plus seasonal cycle
showed in fact that the radiative time constants were long enough
to explain the observed asymmetry. However, latitudinal redistribution
of heat by dynamics would reduce instead of increase the asymmetry
\cite[]{Hour:95b}.
Based on radiative computations, it was then
suggested that the latitudinal variations in atmospheric composition
and the north-south asymmetry in the haze distribution  
could cause the observed asymmetry \cite[]{Beza:95}.
The results obtained with the coupled and uncoupled models 
favor the second explanation.
The coupling with dynamics, and the enrichment
in haze and chemical compounds in the winter hemisphere, clearly contribute
to the seasonal lag in atmospheric temperatures.
However, the model still slightly underestimates the temperature at
southern latitudes.  So some additional dynamical or
radiative effects, not
well resolved by the model, can not be completely ruled out.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{On Polar Enrichment}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\begin{figure}
\centerline{
\includegraphics[height=8cm]{\FIGURES/hcnz.eps}
\includegraphics[height=8cm]{\FIGURES/ktrack.eps}
}
\caption{Meridional transport of HCN during northern winter.
{\bf a:} Meridional cross section of HCN concentration (in ppmv)
(shaded) with stream lines of the mean meridional circulation
(black arrows). {\bf b:} zonal wind (shaded) and
latitudinal eddy diffusivity $K_y$
(contour, 10$^{-6}$~m$^2$~s$^{-1}$). With these particular units,
the same contours can be interpreted as well as iso-values of a meridional
eddy velocity in m~s$^{-1}$ for a mixing length of 1000~km.
 All the quantities are averaged from northern winter solstice
to northern spring equinox.
\label{fg:fig1}} 
\end{figure}

The rationale behind enrichment in most chemical compounds in northern
high latitudes at the time of
the Voyager encounter (close to northern spring equinox) is as follows,
and is illustrated in Fig.~\ref{fg:fig1}.
During almost half a year around solstice,
the mean meridional circulation
is dominated by a global pole-to-pole Hadley cell with the rising branch
in the summer (southern) hemisphere and subsiding motions in the winter
(northern) latitudes (arrows in panel {\bf a} of Fig.~\ref{fg:fig1}).
The downward advection of chemical compounds
from the production zone in the upper stratosphere or mesosphere
is responsible for the strong polar enrichment \cite[]{Lebo:01}
as illustrated for HCN in panel {\bf a} of Fig.~\ref{fg:fig1} (shaded).
This Hadley circulation also creates an intense circumpolar jet in northern
high latitudes (shaded in panel {\bf b} of Fig.~\ref{fg:fig1}).
This jet is unstable on its equatorward flank leading 
the development of planetary waves. Those waves, parametrized in the model
\cite[]{Luz:03II}, are responsible for the erosion of the polar bulge.
Values of the parametrized eddy diffusivity $K_y$ averaged for northern winter are shown in
panel {\bf b} of Fig.~\ref{fg:fig1} (contours).
Figure~\ref{fg:schema} summarizes these mechanisms.

\subsection{Composition of the Polar Vortex}

\begin{figure}
\centerline{
\includegraphics[height=7cm]{\FIGURES/pole.eps}
}
\caption{
A sketch of the essential mechanisms responsible for the observed
contrasts in Titan's stratospheric composition.\label{fg:schema}}
\end{figure}

\begin{figure*}
\includegraphics[width=5.cm]{\FIGURES/tauA1C1Kiid.eps}
\includegraphics[width=5.cm]{\FIGURES/wtauA1C1Kiid.eps}
\includegraphics[width=5.cm]{\FIGURES/hcnA1C1Kiid.eps}
\caption{Time constants and vertical profiles in the winter
polar vortex. {\bf a:} time constants (in Titan days) for
vertical advection in the core of the vortex
($-h/w$, with $h=150$~km, averaged between 70 and 90N), and
meridional circulation ($l/v$, with $l=$1000~km, averaged between 50 and 80N)
and latitudinal mixing by waves ($l^2/K_y$, averaged between 50 and 80N)
at the edge of the vortex. The grey curve corresponds to meridional advection
toward the equator ($-v$ instead of $v$).
The vertical lines correspond to 1~day and 1~year.
{\bf b:} associated scale height (km) for meridional advection, $-wl/v$,
latitudinal mixing $-wl^2/K_y$ and the total scale height
$H=-w/(v/l+K_y/l^2)$.
{\bf c:} vertical profiles of the mole fraction of HCN
(in log scale) at different latitudes (linearized chemistry).
The thin oblique line corresponds to a constant scale height of 100~km.
\label{fg:scaleh}}
\end{figure*}

For the simulations with linearized chemistry, the time evolution of the 
mass concentration $c$ of a given chemical compound is simply given by
\begin{eqnarray}
\frac{\partial c}{\partial t}+
v\frac{\partial c}{a\partial \phi}+
w\frac{\partial c}{\partial z}=\nonumber\\
\frac{c_0-c}{\tau_{\mbox{chem.}}}
+\frac{1}{a^2\rho\cos\phi}\frac{\partial }{\partial \phi}
\left( \cos\phi\rho K_y \frac{\partial c}{\partial \phi} \right)
\label{EQ1}
\end{eqnarray}
where $\phi$ is the latitude, $z$ is the altitude,
$\rho$ is the air density, $a$ is the radius of Titan,
$v$ and $w$ are the meridional and vertical wind components,
$K_y$ is the lateral eddy diffusivity defined above,
$c_0$ is a 1-dimensional vertical profile determined from a 1-dimensional simulation
of Titan's photochemistry and $\tau_{\mbox{chem.}}$ is a time constant from
the same simulation which also depends on altitude only \cite[]{Lebo:01}.

For chemical species subsiding in the polar vortex down from the production
zone, the concentration decreases under the
combined effects of meridional advection of clear air from low latitudes
and latitudinal mixing by waves.
By denoting by $c_{\mbox{eq}}$ the
typical concentration at low latitudes,
the erosion of the polar vortex concentration $c$ by latitudinal eddy diffusion
(last term in equation~\ref{EQ1})
%\begin{equation}
%\frac{1}{a^2\rho\cos\phi}\frac{\partial }{\partial \phi}
%\left( \cos\phi\rho K_y \frac{\partial c}{\partial \phi} \right)
%\end{equation}
can be roughly estimated as
\begin{equation}
\frac{K_y}{l^2} \left( c_{\mbox{eq}}-c\right)
\end{equation}
where $l\simeq$~1000~km  is the typical length scale associated with
latitudinal contrasts in atmospheric composition.
Similarly, the effect on the polar vortex concentration $c$ of
advection of clear air from low latitudes by a meridional
wind $v$ (assumed positive in the northern hemisphere),
$(-v/a) \partial c / \partial \phi$,
%\begin{equation}
%-v\frac{\partial c}{a\partial \phi}
%\end{equation} 
can be approximated as
\begin{equation}
\frac{v}{l}
 \left( c_{\mbox{eq}}-c\right) \mbox{.}
\end{equation} 
Assuming a steady state regime,
the transport equation can be approximated as
\begin{equation}
w\frac{\partial c}{\partial z}\simeq\frac{c_0-c}{\tau_{\mbox{chem.}}}+
\left( c_{\mbox{eq}}-c\right) \left( \frac{K_y}{l^2}+\frac{v}{l} \right)
\end{equation}
For a large number of species, the polar enrichment is such that the
hypothesis that $c_0 << c$ and $c_{\mbox{eq}} << c$ can be made.
In such conditions, and for a positive meridional wind $v>0$,
the vertical gradient of $c$ in the polar vortex
is given by 
\begin{equation}
w\frac{\partial c}{\partial z}\simeq
-c\left( \frac{1}{\tau_{\mbox{chem.}}}+\frac{K_y}{l^2}+\frac{v}{l} \right)
\end{equation}
corresponding to a total atmospheric scale height for  the mass concentration of
\begin{equation}
H=\left( \frac{d \ln c }{dz} \right)^{-1}\simeq 
w\left( \frac{1}{\tau_{\mbox{chem.}}}+\frac{K_y}{l^2}+\frac{v}{l} \right)^{-1}
\end{equation}
For species with large chemical time constants (compared to dynamical
time constants), the shape of the vertical concentration profile is controlled
by a competition between vertical advection (with faster advection from the
upper stratosphere reducing vertical contrasts) and relaxation toward
the weak concentrations found in low latitudes in the mid stratosphere 
by either mean meridional transport or latitudinal mixing by waves
(which accentuates the vertical contrasts).
The relative importance of meridional advection and
latitudinal mixing can be quantified
either in terms of time constants,
$\tau_{\mbox{mmc}}=l/v$ and $\tau_{\mbox{mix}}=l^2/K_y$
or in terms of the associated scale height $-w\tau_{\mbox{mmc}}$ and $-w\tau_{\mbox{mix}}$.




In Fig.~\ref{fg:scaleh}, panel {\bf a}, we show for northern
winter (quantities averaged between northern winter solstice and
northern spring equinox) the time constants for
the vertical and meridional advection as well as for latitudinal mixing by 
waves.
The meridional advection dominates latitudinal transport only
in the upper branch of the Hadley cell (above 400~km) as well
as in the lower stratosphere, between 100 and 150~km were 
the meridional wind is negative. This region corresponds to the lower
branch of the stratospheric Hadley cell.
Typical values for dynamical time constants vary from about one Titan year at
120~km to less than one Titan day above 350~km.
As discussed below, those time constants are shorter
or much shorter than chemical time constants for most species.
The associated dynamical scale height (panel {\bf b})
is of the order of 100~km in the mid-stratosphere,
where it is essentially determined by the balance between downward
advection and latitudinal mixing by waves.

Panel {\bf c} shows, at the same season, simulated vertical profiles of HCN 
(obtained with the linearized chemistry)
for different latitude ranges in the northern hemisphere.
The straight line in the same panel corresponds to a scale height of 100~km.
This is very close to the scale height for HCN profiles
above 250~km at 45-60N,
at the edge of the polar vortex, in the region of mixing.
Between 100 and 150~km, the scale height is much larger due to the weaker
mixing there. For the 45-60N profile, a local maximum is even obtained
in this altitude range, due to meridional advection of rich air
from the polar vortex.

The shape of the vertical profiles, with three distinct slopes and
a very well mixed region in the mid-stratosphere, between 150 and
400~km, is consistent with
recent observations by \cite{Mart:02}.
Numerical simulations with 1-dimensional chemical models show much larger
vertical variations in this region due to a much weaker vertical transport
\cite[see e.g. Figure~1 of ][]{Hida:02}.
The vertical advection by the stratospheric Hadley
cell is in fact much more efficient in transporting chemical species
over long distances than turbulent mixing by small scale eddies.
For a vertical eddy diffusivity $K_z$ and a vertical scale $h$,
the time scale for diffusive mixing
is  $h^2/K_z$  instead of $h/w$ for advection.
Between 250 and 150~km, the time constant associated with vertical advection
varies from about 10 to 100~Titan days.
Very large values of $K_z$ of typically 100-5000~m$^2$~s$^{-1}$
would be required (assuming $h\simeq$200~km) to obtain similar time
constants.
This is consistent with the study by \cite{Lebo:03a} in which an eddy diffusion coefficient of 
100~m$^2$~s$^{-1}$ was adopted in this region to emulate mixing by the
general circulation.
% h^2=2.25e10 / tau=10*16*86400 1.3e7  =  1.7e2

\subsection{Magnitude of Latitudinal Contrasts}

\begin{figure}
\includegraphics[angle=-90,width=8cm]{\FIGURES/compo.eps}
\caption{Latitudinal profiles of chemical species as computed with
the 4 options (full, linearized or null chemistry as well as idealized
tracers) compared to Voyager observations.
Curves are normalized to the observed equatorial value to focus
on latitudinal variations.\label{fg:chim}}
\end{figure}


\begin{table*}
\caption{Condensation level, polar enrichment
(ratio of the averaged concentrations at 50-70N and 0-30N)
and chemical time constants for various chemical compounds.
\label{tab:chem}}
\begin{center}
\begin{tabular}{ccccccccc}
                    & C$_2$H$_2$ & C$_2$H$_4$ & C$_2$H$_6$ & CH$_3$CCH & 
                      C$_3$H$_8$ & C$_4$H$_2$ &  HCN & HC$_3$N \\\hline 
\multicolumn{9}{c}{Condensation level}\\\hline
winter polar vortex, km  & 64 &   -   &  57 &  64 & 54 &  77 &  86 & 63 \\
winter polar vortex, hPa & 35 &   -   &  50 &  35 & 60 &  20 &  15 & 36 \\
retained for idealized tracers, hPa    & 40 &   -   &  80 &  40 & 100 & 20 &  20 & 40 \\\hline
% niveaux detectes a partir du script levels.x sur SEB
%winter polar vortex & 28   & -    & 36   & 28     &28   & 16   &  12  & 28 \\
%idealized tracers   & 40   & 200  & 60   & 40     &40   & 30   &  20  & 40 \\\hline
\multicolumn{9}{c}{Level of maximum of the weighting function for Voyager IRIS observations}\\\hline 
km        & 152  & 190  & 160 &  85   &125  & 100  &  107 & 103  \\
hPa       & 2.5  & 1    & 2   &  13   &5    & 10   &  8   & 9  \\\hline %SEB
\multicolumn{9}{c}{Polar enrichment}\\\hline
Observations &
2.1 &
8.3 &
1.4 &
6.4 &
1.9 &
16.8 &
7.6 &
19.6 \\
     Full chemistry & 
     2.7 &
     3 &
     1.9 &
     9.9 &
     1.6 &
     5.9 &
     7.8 &
     6.2 \\
     Linearized chemistry & 
     2.3 &
      - &
      1.3 &
       - &
        - &
	 - &
	 7.5 &
	  - \\
	  Null chemistry & 
	  2.6 &
	  2.6 &
	  1.9 &
	  9.6 &
	  1 &
	  10.1 &
	  9.2 &
	  5.2 \\
	  Idealized tracers & 
	  2.4 &
          - & 
	  2 &
	  5.6 &
	  2.6 &
	  9.3 &
	  9.3 &
	  4.2\\\hline
\multicolumn{9}{c}{Chemical time constant (Titan year)}\\\hline
0.5-1 hPa (190-200~km)         & 0.5-100 & 0.03-4 & $>$40  & 2-100 & $>$10 & 0.001-1 & 2-200 & 0.05-10 \\
2-5 hPa  (125-160~km)         & $>$2.5  & 0.2-5  & $>$200 & 2-200 & $>$80 & 0.01-3  & 2-300 & 0.2-50 \\\hline
\end{tabular}
\end{center}
\end{table*}

\begin{table*}
\caption{Characteristics of the various chemical schemes.\label{tb:sims}}
\begin{center}
\begin{tabular}{|c|c|c|c|}\hline
Simulation name      & Chemistry & condensation & upper boundary condition \\\hline
Full chemistry & realistic & realistic & prescribed flux \\
Null chemistry & no        & realistic & prescribed flux \\
Linearized chemistry & Relaxation & realistic & relaxation \\
Idealized tracers & no     & removal below a given pressure level & relaxation\\\hline
\end{tabular}
\end{center}
\end{table*}

The enriched subsiding air generally reaches saturation when approaching
the tropopause level, explaining the sharp decrease observed in 
panel {\bf c}\ of Fig.~\ref{fg:scaleh}
for HCN at all latitudes.
For most species, the concentration at saturation at the very cold tropopause
of Titan ($T\simeq70$~K) is orders of magnitude smaller than the
concentrations observed in the polar vortex.
So the air which is recycled in low latitudes
rises from the troposphere with a very low concentration.
It is then replenished either by latitudinal advection from
the polar vortex or by latitudinal mixing by waves.
The latitudinal contrast observed for altitudes ranging from 80 to 200 km
thus essentially represents the efficiency of mixing and latitudinal transport
to carry chemical species into the clear air rising from the troposphere.
As is visible in Fig.~\ref{fg:scaleh}, meridional advection dominates this latitudinal
transport below 150~km (grey curve in panel {\bf a}).

This simple idea of the build-up of latitudinal contrasts was checked
by simulating chemical compounds as very simple atmospheric tracers
(thereafter named  idealized tracers). 
In the three upper layers, those tracers are relaxed (with the same time
constants as for the linearized chemistry) toward a concentration
of unity.
The tracers undergo both the mean meridional circulation and
latitudinal mixing, but not chemistry. The tracers are completely removed
below a given level, close to the tropopause.
The various tracers only differ by the pressure level at which they are
removed.
We consider 8 idealized tracers corresponding to condensation levels of:
100., 80., 60., 40., 30., 20., 15. and 10. hPa.
The idealized tracers can only be compared to real chemical compounds in terms
of relative latitudinal or vertical variations.
Also, because the idealized tracers disappear suddenly  at the condensation
level whereas the actual concentration is decreasing gradually in the
condensation region, the level retained for the condensation
of the idealized tracer must be somewhat lower than the real condensation
level of the chemical compound to which it is compared
(typically at the level
where the concentration has decreased by one order of magnitude, which
generally occurs at a little bit less than twice the pressure of
the actual condensation level).
The actual level of condensation in the winter polar vortex was computed
from the simulation with full chemistry, as the upper layer where the
given species condenses (note that this level depends on the concentration
simulated for that species in the polar vortex).
This level as well as the
level retained for the idealized tracer is given for a certain number of species
in Table~\ref{tab:chem}.


Table~\ref{tab:chem} also shows the polar enrichment, defined as the
ratio of the averaged concentrations at 50-70N and 0-30N.
First, the comparison of the observed polar enrichment with the
results of the model with full chemistry 
shows an agreement for most species, except C$_2$H$_4$ -- the 
enrichment appears closer to the pole than observed -- C$_4$H$_2$ 
and HC$_3$N -- which may be affected by chemical processes not included
in the model (e.g. pathways producing haze precursors).
For most species (C$_2$H$_2$, C$_2$H$_6$, C$_3$H$_8$, HCN, HC$_3$N), 
the polar enrichment obtained for the full chemistry
and idealized tracers differ by less than 50$\%$.
This general agreement
clearly shows that the level at which species are removed by condensation
is controlling for a large part the magnitude of polar enrichment.
The secondary role played by either chemistry or specifications of
the model upper boundary conditions is detailed below.

In order to estimate the relative importance of chemistry for
the various species, we computed a chemical time constant
$\left( d\mbox{ln}c/dt \right)^{-1}$,
where the tendency is computed for all the chemical reactions acting as
a sink for the given species. Values are given for two pressure ranges
in  Table~\ref{tab:chem}.
We also ran the full chemical model but replacing 
all chemical rate constants by zero (thereafter named null chemistry,
see Table~\ref{tb:sims}).
Values of the polar enrichment obtained by the full and null chemistry
depart by less than 20\% except for C$_4$H$_2$ and C$_3$H$_8$.
The disagreement between null and full chemistry for C$_4$H$_2$  is
consistent with the relatively short chemical time constants obtained
for this species.
In the case of C$_3$H$_8$, the discrepancy between full and null chemistry,
despite very long chemical time constants, is probably due
to the fact that the flux at the upper boundary is very close to zero.
This flux is probably underestimated by the 1-dimensional
photochemical model, leading to an underestimated equatorial value in the GCM.
The photochemical reactions,
though occurring on long timescales, may then explain the difference between
both simulations (Table~\ref{tab:chem}).
As a consequence also, the simulation of C$_3$H$_8$ as an idealized
tracer, with an upper condition prescribed as a relaxation 
toward a fixed value, also departs from the null chemistry.

For the other species, the agreement between the null
chemistry and idealized tracer is very good, in spite of the different
treatment of the upper boundary condition and of the condensation
(see summary of the different simulations in Table~\ref{tb:sims}).
This good agreement reinforces the confidence in the results presented
above.
In particular, the upper boundary condition which, for a given species,
controls its mean level in the lower stratosphere, does not affect
its latitudinal variations.

Finally, it clearly appears that the level of condensation is the
main determinant of the magnitude of polar enrichment for most species.
The higher the condensation occurs the stronger the polar enrichment.
If the observational contribution function peaks just above the 
condensation level, the
observed polar enrichment will also be larger than if the contribution
function peaks much higher in the atmosphere.
This for instance explains the small polar enrichment of C$_2$H$_2$ when
compared to CH$_3$CCH  which both condensate at the same levels.

The same processes explain not only the magnitude of the polar
enrichment but also the shape of the latitudinal profiles.
To illustrate this point,
we show in Fig.~\ref{fg:chim} for the three radiatively active species 
the latitudinal profiles normalized to the observed equatorial value for the
various simulations.
We compare normalized rather than absolute profiles to focus on latitudinal
variations and also because comparison of absolute values does not make sense
for the idealized tracers which are arbitrarily relaxed toward a concentration
of unity at the model top (comparisons of absolute profiles are shown in
Fig.~\ref{fg:compo} for the full and linearized chemistry).
The agreement between all those profiles, from the most complex
with full chemistry to the simple idealized tracer, clearly confirms
the very weak control by chemistry of those latitudinal variations.



%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Constraints on Winds}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\begin{figure}
\centerline{\includegraphics[angle=-90,width=8cm]{\FIGURES/chim.eps}}
\centerline{\includegraphics[angle=-90,width=8cm]{\FIGURES/T.eps}}
\centerline{\includegraphics[angle=-90,width=8cm]{\FIGURES/u.eps}}
\caption{Sensitivity experiments.
Polar enrichment
in C$_2$H$_2$ and HCN obtained with the full coupled model, 
or with uncoupled  chemistry (No chem) or uncoupled haze (No haze) 
or both (uncoupled) or with weakened (0.3xKy) or enhanced (3xKy) dissipation
by waves or modified advection field for chemical compounds (mean meridional
circulation multiplied by 0.3 or 3).
The error bars correspond to the Voyager data.
\label{fg:sensitiv}}
\end{figure}

\begin{figure}
\centerline{\includegraphics[width=9cm]{\FIGURES/couple.eps}}
\caption{Effect of the coupling with the haze on the latitude-altitude distribution
of HCN ({\bf a}) and C$_2$H$_2$ ({\bf b})
and associated impact on the mean meridional
circulation (MMC, {\bf c}) and latitudinal eddy diffusivity $K_y$ 
({\bf d}, 10$^{-6}$~m$^2$~s$^{-1}$)
at the time of the Voyager encounter.
\label{fg:couple}}
\end{figure}

If the latitudinal contrasts depend primarily on dynamics as well
as on the level of removal by condensation for each species,
what does observation of the composition tells us about winds?
Or, to which degree is the agreement with observations sensitive
to the simulated circulation?

To answer that question, we made a series of sensitivity experiments
by varying the intensity of either the meridional advection or 
latitudinal mixing.
A factor was applied either on  both  $v$ and $w$ or on 
the eddy diffusivity $K_y$, with two values~: 0.3 and 3.
The factor was applied for the transport of the three linear chemical species
only.

We also performed simulations with and without coupling with chemistry
or haze microphysics.
The coupling with haze is turned off by applying a strong latitudinal
mixing on haze particles to obtain almost no latitudinal contrasts.
In those simulations, the production rate of haze must be strongly
reduced to obtain the same geometric albedo of Titan
\cite[see ][for additional details]{Rann:04}.
In the radiative computation, coupling with chemistry is turned off 
by using the \cite{Lell:89} vertical profile of the methane abundance, 
and values taken from \cite{Cous:89} for ethane and acetylene above their
condensation level.
In this simulation however, the chemical
compounds are transported as in the coupled version.
Note that, to avoid complex feedbacks which would complicate the analysis, 
the simulations with modified meridional circulation or latitudinal mixing
are performed without coupling with chemistry. 

In Fig.~\ref{fg:sensitiv}, we compare the polar enrichment for C$_2$H$_2$
and HCN obtained in the various sensitivity experiments and in 
Voyager observations (with associated error bars).
For the nominal version, the latitudinal contrast is very well reproduced
for both species. The ratio is much larger (respectively weaker) if 
the meridional advection ($v$ and $w$ components)
is increased (respectively decreased) or if
the latitudinal mixing is weakened (respectively strengthened).
So observations of the composition actually contain 
strong constraints on the combined intensity of advection and mixing.
The results suggest that those components are predicted at a factor of
typically less than two with the nominal version of our coupled Titan
GCM.

Coupling with chemistry tends to degrade a little bit
the agreement but keeps the results in the error bars.

Without haze coupling, the agreement is significantly reduced, 
the contrast for HCN being much too low and that of C$_2$H$_2$ too high. 
The difference in sensitivity of these two species is due
to the level at which they are observed.
In Fig.~\ref{fg:couple}, we show a latitude-altitude cross-section
of the ratio of the concentrations
of HCN (panel {\bf a}) and C$_2$H$_2$ (panel {\bf b}) in
the coupled and uncoupled simulations at the time Voyager encounter.
The sensitivity is in fact very similar for the two species, with
a decrease by about 30$\%$ in the polar vortex near 160~km
and an increase by about 50$\%$ near 100~km, and can be interpreted
as follows.

The main effect of coupling with haze is a reinforcement
of the subsidence in the winter polar region, due to a larger infrared
cooling to space due to the accumulation of haze in the polar
night \cite[]{Rann:04}.
This reinforcement of the meridional circulation is illustrated by
the stream-lines in panel {\bf c} of Fig.~\ref{fg:couple}. It is responsible
for the increase in concentration in the low stratosphere, near 100~km.
At the same time, the stronger jet in the coupled version leads to
stronger wave activity (panel {\bf d}).
The latitudinal eddy diffusivity is much larger in the coupled version.
This reinforcement is particularly strong near 150~km and explains
the weaker polar enrichment in this altitude range in 
the coupled model.


C$_2$H$_2$ which is observed at 2.5~hPa, close to 160~km, has a smaller
polar enrichment in the coupled version while it is the opposite for HCN
which is observed close to 100~km.



 
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Seasonal Variations}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\begin{figure*}
\centerline{\includegraphics[angle=-90,width=18cm]{\FIGURES/season.eps}}
\caption{The seasonal evolution of 
the concentration of HCN  (ppmv) at (a) 50-70N and (b) 0-30N 
as well as the corresponding polar enrichment (c) are shown, together
with the various  components of atmospheric transport:
vertical wind in the polar vortex (d) and meridional wind (e)
and latitudinal mixing (f) at the edge of the polar vortex.
WS is for winter solstice, SE for spring equinox, SS for summer solstice and
FE for fall equinox. 
\label{fg:season}}
\end{figure*}

\begin{figure*}
\centerline{\includegraphics[width=18cm]{\FIGURES/seasonhcn.eps}}
\caption{ 
\label{fg:clouds}
Six snapshots of the meridional structure for HCN and associated mean %SEB
meridional circulation
for seasons going from northern fall to northern spring equinox.
Seasons are labeled according to the solar longitude $L_s$ with
$L_s$=180 for northern fall equinox, $L_s=270$ for northern winter solstice
and $L_s=0$ for northern spring equinox.
Concentration of HCN (grey scales) is in ppmv. The condensation rate of
HCN is shown as shaded squares with the same bar chart but for
$10^{9}$ m$^{-3}$ s$^{-1}$.}
\end{figure*}

We have focused up to now on the mean winter conditions for interpreting
the physics of the transport of chemical compounds and on 
northern spring equinox for the comparison with Voyager observations.

In Fig.~\ref{fg:season} we present how the northern polar enrichment in HCN,
as well as the associated components of atmospheric transport, evolve
with seasons in the mid stratosphere (80-240~km).

Soon after fall equinox in this altitude range,
the high-latitude concentration grows very fast under the
effect of downward advection (panel {\bf a}).
At the same time, the jet is not yet
established so that the latitudinal contrasts with low latitudes
grow very fast, reaching a first maximum (panel {\bf c}).
The polar enrichment then decreases under the combined
effect of mixing by waves and latitudinal advection from the vortex toward
low latitudes.
Close to northern spring equinox, the meridional advection
reverses above 150~km. The chemically rich air of the polar vortex
is then rapidly advected toward the other
hemisphere, explaining both the rapid decrease of concentrations at 50-70N
above 180~km (panel {\bf a}) and the local maximum observed at
0-30N, in the upper part of the domain of panel {\bf b}.
In the lower part, the polar high concentrations persist much longer,
both because the time constants are longer there, and because the vertical
and meridional advection do not change their signs in high latitudes.

Around equinox on the other hand, 
the ascending branch of the stratospheric Hadley cell
migrates from the hemisphere coming out of summer
to the other hemisphere, bringing clear tropospheric
air to the lower stratosphere at low latitudes. This upward advection
of clear air explains the decrease observed below 150~km at 0-30N
(panel {\bf b}) and the associated maximum in polar enrichment (panel {\bf c}).
At this season also, the intensity of latitudinal mixing
reaches a minimum in the lower stratosphere.
But, as already stated, it is essentially the mean meridional advection
which controls the polar enrichment below 150~km (compare panel {\bf e}\
and {\bf f}).
The time constant of about 5 terrestrial years which
can be deduced from the decrease of the first maximum of polar enrichment
(panel {\bf c}) is also the time it takes to travel
1000~km at a speed of 5~mm~s$^{-1}$, which is the typical meridional
velocity encountered in the lower branch of the stratospheric Hadley cell,
near 120~km (panel {\bf e}).

This seasonal evolution is illustrated in a series of 6 snapshots running 
from northern fall to northern  spring equinox (Fig.~\ref{fg:clouds}). 
At both equinoxes (first and last panels) the upward advection of
clear air by the low latitude branch of the two equator-to-pole
Hadley cells is clearly visible. At fall equinox (panel {\bf a}),
the southern polar vortex is still strongly enriched while the
rich air from the upper stratosphere starts to sink in the 
northern high latitudes. Then the global pole-to-pole Hadley
cell sets in (panels {\bf b} to {\bf d}), with an ascending branch in the south and subsidence 
in the north. Between 80 and 200~km, a secondary Hadley cell 
appears in the summer hemisphere which persists until the next equinox. 
This secondary cell clearly contributes to maintain a strong enrichment
in the summer hemisphere, in the lower stratosphere.

Note that this secondary cell is much more intense in the simulations
with coupling to both atmospheric composition and haze.

Because of this secondary cell also, the air keeps on sinking into
the troposphere during summer, producing condensation at high
latitudes (condensation is shown as shaded squares close to 80~km
altitude in Fig.~\ref{fg:clouds}), even close to summer solstice when
insolation is maximum at the pole. Exactly the same picture
is obtained with C$_2$H$_6$ and C$_2$H$_2$ (not shown).
The secondary cell coupled to condensation is in fact the
sink of polar enrichment of the lower stratosphere during summer.
This may also explain the cloudy features observed
around the pole during summer from ground based observations 
\cite[]{Grif:98,Grif:00,Brow:02,Roe:02} and more recently from the T0 Cassini flyby of Titan.
Our results clearly show  that there is no contradiction
between the season (summer) and a stratospheric origin of the observed
clouds,  so that a convective origin from the troposphere is not
required.
From our computation, it appears that clouds are as likely
in the summer as in the winter hemisphere.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{What should Cassini Observe?}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\begin{figure}
\includegraphics[width=8cm]{\FIGURES/cassini.eps}
\caption{
Time evolution of some observable quantities. {\bf a}:
polar enrichment for HCN (at an altitude of 107~km) and
C$_2$H$_2$ (at 152~km). {\bf b}: zonal wind at the equator
(20S-20N) and in the northern high latitude jet (40-70N) at an altitude
of 250~km. {\bf c}:
Equator to high latitude (60N) temperature contrast
and north south asymmetry (temperature difference between 60N and 60S)
at 190~km.
\label{fg:cassini}
}
\end{figure}
Most that is known about the composition and dynamical state of
Titan's mid stratosphere (pressure range 10-0.1 hPa) comes from the
Voyager observations, at northern spring equinox.  The vertical
temperature profile, from the surface up to 200~km was reconstructed
from radio occultation measurements \cite[]{Lind:83}.  From there
upward it is mainly constrained by the model by \cite{Yell:91}.  For
stratospheric dynamics, an estimate of the zonal flow was inferred
through geostrophic balance \cite[]{Flas:81} from the temperatures
retrieved for two pressure levels (0.4 and 1~hPa) from observations of
the Q and P branches of the $\nu_4$ band of methane by the Voyager
Infrared Spectrometer (IRIS).  Since then, ground-based observers have
tried to measure zonal winds with two main techniques: stellar
occultations \cite[]{Hubb:93,Bouc:03}, which provide the speed but
leave the wind direction undetermined, and from the Doppler shifts of
either ethane and nitrile emission lines \cite[]{kostiuk2001,moreno2003} or
of the backscattered solar spectrum \cite[]{luzdps2003}.

The stratospheric composition is also constrained from Voyager IRIS
observations. The equatorial composition has been obtained in the
region 80-200 km for several hydrocarbons and nitriles (C$_2$H$_2$,
C$_2$H$_4$, C$_2$H$_6$, CH$_3$C$_2$H, C$_3$H$_8$, C$_4$H$_2$, HCN),
and latitudinal variations of their abundances could be retrieved
thanks to the latitudinal resolution of IRIS \cite[]{Cous:89,Cous:95}.
Ground-based observations made since only give access to a mean
stratospheric value (C$_2$H$_6$ \cite[]{Live:02}; C$_3$H$_8$
\cite[]{Roe:03}) or to a mean stratospheric vertical profile (HCN,
CH$_3$CN, HC$_3$N \cite[]{Mart:02}), which as mentioned above contains
also information on atmospheric dynamics.

Latitudinal contrasts in haze also contain information on
dynamics. They have been monitored by Voyager \cite[]{Srom:81} and,
since then, by the Hubble Space Telescope
\cite[]{Lore:99,Lore:01,Youn:02} and ground-based observatories with
adaptive optics and speckle imaging
\cite[]{RoeN:02,Gend:04,Gibb:04}. Transient cloud features have been
detected from disk-integrated infrared spectra \cite[]{Grif:00}, and
from resolved adaptive optics \cite[]{Brow:02,Roe:02} and speckle
imaging \cite[]{Gibb:04}. Haze aspects of our simulations are already
published \cite[]{Rann:02,Rann:04}. Although strongly coupled to
dynamics, the distribution of haze and clouds are much more sensitive
to the way the microphysics is handled than chemical composition.
Thus constraints on the dynamics are less direct than for chemical
compounds.

The Cassini orbiter has been placed in Saturn's orbit on the 1st of
July 2004.  During four years it will make several Titan fly-bys,
which will be used to modify the orbits around Saturn. It will monitor
at least a part of the seasonal cycle. Information on the dynamics of
the mid-stratosphere will be provided essentially by observations of
thermal contrasts by the composite infrared spectrometer (CIRS), but
recent observations by the Imaging Science Subsystem (ISS) indicate
that cloud tracking may become a possibility. The Doppler Wind
Experiment (DWE) onboard the Huygens probe will measure the speed and
direction of the winds to better than 1~ms$^{-1}$ accuracy during the
descent \cite[]{Bird:02}, but only below 160 km and in a very limited
time span. Both the Descent Imager/Spectral Radiometer (DISR) and the
Huygens Atmospheric Structure Instrument (HASI) data sets will also
allow to corroborate DWE measurements \cite[]{Alli:04,Fulc:02}.

Here we propose some predictions of what we expect Cassini-Huygens
to observe in Titan's mid stratosphere.
Of course the model could predict a number of things.
In Fig.~\ref{fg:cassini}, we show predictions concerning
quantities which were
observed by Voyager but at a different season.
We first show the seasonal evolution of polar enrichment for 
HCN and C$_2$H$_2$, for altitudes corresponding to the maximum of
the weighting function of Voyager infrared retrievals
(107 and 152~km respectively).
The two-peak maximum described above is clearly visible for
the polar enrichment of HCN (panel {\bf a}).
For  C$_2$H$_2$, the second maximum is not visible
not because of their different condensation heights,
but rather because of the higher altitude probed.
Note also that the enrichment for HCN increases with a phase lag of 1-3 terrestrial
years with respect to C$_2$H$_2$, at the beginning of fall.
This phase lag is directly related to the difference in altitude
of the two weighting functions (50~km lower for HCN).
The corresponding downward velocity in the polar vortex is of the
order of 0.5-1.5~mm~s$^{-1}$, which is consistent with the values given
in Fig.~\ref{fg:season}. Here, the vertical wind can be deduced almost
directly from the observation of the atmospheric composition.

In the second panel, we show the time evolution of the stratospheric
jet speed (40-70N) and equatorial zonal wind velocity at 250~km.
It is not clear whether Cassini will succeed in measuring directly the
winds by tracking clouds or other  structures, since, except for recent
ground-based detections of clouds at the south pole, Titan's disk
seems featureless. However, the 250~km-altitude range is interesting
because it can be probed by stellar occultations (including the 2003
occultation, B. Sicardy, priv. comm.) and by heterodyne infrared
spectroscopy \cite[]{kostiuk2001}.
The time evolution of the stratospheric jet presents similarities
with that of HCN, with a two-maxima structure and a long
winter season with an intense circumpolar jet, contrasting with a shorter 
season where the wind speed is less than the equatorial velocity.


The time evolution of the latitudinal temperature contrast at 
1~hPa or 190~km, which will be easily monitored by Cassini, is still more 
complex. The polar latitudes are generally colder than the equator 
except during a short season around the summer solstice.

The curves shown in Fig.~\ref{fg:cassini}
are relative to the northern hemisphere but the situation
is, to first order, symmetric for the southern hemisphere
(not exactly because of the eccentricity of Titan's orbit around the Sun)
so that
the situation at the time of arrival is given by year 2005 for
the northern hemisphere and 2020 for the southern hemisphere.
According to those figures, Cassini should arrive at the time at which 
the zonal winds are weakest in the southern hemisphere, with
southern high latitudes warmer than the equator.

As shown in panel {\bf f}\ of Fig.~\ref{fg:season}, it should be a period of
intense wave activity.  From Fig.~\ref{fg:scaleh}, it is seen that the time
constants of latitudinal mixing and vertical advection are of the
order of tens of Titan days in the mid stratosphere. According to the
simulations of latitudinal mixing by planetary waves presented by
\cite{Luz:03I}, longitudinal contrasts less than 10\% of the
latitudinal contrasts are expected, which could be monitored as well
by Cassini's CIRS.

Many other quantities will be observed by remote sounding by Cassini
and {\em in situ} measurements by the Huygens descent module. A
database based on a reference simulation performed with the coupled
model is now distributed to the scientific
community \footnote{http://www.lmd.jussieu.fr/titanDbase}, so that it
will be possible to check a posteriori if the observations confirm or
not other components of the simulations.

\section{Concluding Remarks}

The results presented above have been obtained with a
state-of-the-art comprehensive climate model of Titan,
including coupling with both chemistry and haze microphysics.
This model, however, presents a number of approximations and limitations
which must be kept in mind when discussing the  results.
First, the model is reduced to an axi-symmetric  version. Latitudinal mixing by waves is handled through a parametrization 
which obviously only partially and imperfectly reproduces
the physics of these waves. 
In particular, there was no attempt to predict mixing by baroclinic waves
which may become significant at high latitudes.
Also, because of the 2-dimensional nature of the model as well as of
the lack of information on
surface conditions, longitudinal forcing, as could occur from the presence
of mountains, is not included.
Subgrid scale gravity waves, which could be parametrized as in  
many terrestrial or Martian GCMs, are not accounted for either, once
again essentially because of the lack of information on the possible
sources. The dynamical core of the model is based on the primitive
equations of meteorology, which assume the thin layer approximation,
whose validity becomes doubtful at  
altitudes as high as 400~km on Titan.
The thin layer approximation could in particular affect the magnitude of
superrotation since the altitude with respect to the surface is not
taken into account when computing the distance to the polar axis which
enters in the definition of angular momentum.
Because of this approximation, an overestimation by 10\% of the zonal
wind can be expected at the peak altitude for superrotation, near 300~km. 
At the model upper boundary also, the hypothesis of Local Thermodynamic
Equilibrium is probably violated \cite[above a few tens of Pa, ][]{Yell:91a}.
As for the composition, the mean stratospheric concentrations strongly depend on
the top boundary conditions and on the chemistry occurring above,
for which we rely on the results of a 1-dimensional photochemical model
(with no latitudinal nor seasonal variations).
The haze production rate is not coupled to the  chemistry, although
some attempts are under way to include a parametrization of polymerization. 
Condensation/sublimation of methane, which has been shown to
significantly affect
the tropospheric circulation \cite[]{Toka:01} is not accounted for.
Nor is the possible effect of gravity tides from Saturn \cite[]{Toka:02}.
Finally, we had to make an assumption on the thermal inertia of the surface.
This thermal inertia was fixed to a low value, typical of continental
surfaces on Earth. However, tests done with a much larger thermal inertia,
typical of an ocean of water, do not change the results, except slightly in the 
lower troposphere (results not shown).


Despite all those limitations, the coupled model provides
a comprehensive and consistent view of the main observations made
by Voyager, and since then by ground based observations, 
concerning the composition and dynamics of the stratosphere.
In particular, the polar enrichment in chemical compounds in the
mid stratosphere caused by downward advection
of air coming from the source region \cite[]{Lebo:01} is
remarkably well reproduced with the nominal version of the model.
For most species, this
polar enrichment has little to do with chemistry.
It is due to the contrast
between  rich air in the polar vortex and uprising clear air having
undergone condensation when approaching the tropopause.
The amplitude of the polar enrichment is controlled by the competition
between vertical advection and latitudinal transport by either
the mean meridional wind or parametrized eddies.
The modulation of the observed polar enrichment
for the various species comes from the different 
altitudes at which they are removed by condensation and from the 
altitude of the peak of the observational contribution function.
The vertical gradient of atmospheric composition in the polar
vortex is also controlled by the balance between vertical
advection and latitudinal mixing with the clear air of 
equatorial latitudes.

Since latitudinal and vertical contrasts are primarily
dependent on atmospheric dynamics,
it also means that observations of the composition provide a strong 
constraint on the stratospheric circulation.
In particular, the good agreement with available observations for
the nominal version 
provides an indirect but strong validation for the atmospheric
regime obtained in the general circulation model (for
the mean meridional circulation as well as for the parametrization
of latitudinal mixing by waves) and in turn for the
explanation put forward for explaining 
Titan's atmospheric superrotation in previous
studies \cite[]{Hour:95b,Luz:03II}.

Note that there is some analogy with the description of humidity contrasts
in the Earth's troposphere. For humidity, the source is at the surface and
the value  at saturation decreases with increasing altitudes up to the
tropopause. In the absence of mixing,
the specific humidity of a given air parcel corresponds to the smallest
humidity at saturation encountered along the parcel trajectory (more or less
corresponding to the highest point).
There also, water vapor can be used to monitor atmospheric transport
and assess transport models  \cite[e.~g.][]{Pier:98}.

It is not clear yet whether Cassini will be able to monitor cloud motions.
It is clear however that the time evolution of stratospheric temperature
and composition will provide a strong constraint on the atmospheric dynamics
and on the validity of the coupled model which we will
continue to develop.


An extended mission beyond the nominal 4 years 
will probably add a lot to our understanding
of the complex machinery of Titan's stratosphere.
For instance, the transition period after equinox will bring substantial 
information on vertical advection in the polar vortex. 
