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Title: Eddy Diffusion Profile of Titan's Atmosphere Revealed by Cassini Observations
Article Type: Special issue: Outer Planet IX
Keywords: Titan; Eddy diffusion profile; Chemical kinetics
Corresponding Author: Mr. Cheng Li,
Corresponding Author's Institution: California institute of technology
First Author: Cheng Li
Order of Authors: Cheng Li; Xi Zhang, PhD.; Joshua A Kammer; Mao-Chang Liang, PhD; Yuk L Yung, PhD
Abstract: Abstract The recent measurements from the limb-view soundings of Cassini/CIRS and the
stellar occultations from Cassini/UVIS revealed the complete vertical profiles of minor species (e.g.,
C2H2 and C2H4) from 100 to 1000 km in the atmosphere of Titan. In this study, we developed an
inversion technique to retrieve the eddy diffusion profile using C2H2 as a tracer species. We find that
the new eddy profile features a low eddy diffusion zone near the altitude of detached haze layer (~550
km), which might lead to the formation of the detached haze layer through a positive feedback process.
The new photochemical model using the retrieved profile are in better agreement with the Cassini
measurements than the previous 1D chemistry-diffusion models . Given the retrieved eddy diffusion
profile, we perform a linear perturbation test to identify key reactions that control the abundance of
C2H4 in the upper stratosphere (~200 km). New expressions of the rate coefficients for two threebody reactions are proposed so as to bring the simulated profile closer to the observation. The revised
rate coefficients should be investigated in the laboratory and applied to the hydrocarbon chemistry of
giant planets.
coverletter.PDF
Eddy Diusion Prole of Titan's Atmosphere Revealed by Cassini
Observations
Cheng Li , Xi Zhang , Mao-Chang Liang , Yuk L. Yung
1,
1
1,1
1
a California Institute of Technology, 1200 E. California Blvd, MC 150-21, Pasadena, CA 91125,
b Research Center for Environmental Changes, Academia Sinica, Taipei, Taiwan
c Graduate Institute of Astronomy, National Central University, Zhongli, Taiwan
USA
Abstract
The recent measurements from the limb-view soundings of Cassini/CIRS and the stellar
occultations from Cassini/UVIS revealed the complete vertical proles of minor species (e.g.,
C H and C H ) from 100 to 1000 km in the atmosphere of Titan. In this study, we developed
2
2
2
4
an inversion technique to retrieve the eddy diusion prole using as a tracer species. We nd
that the new eddy prole features a low eddy diusion zone near the altitude of detached
haze layer (∼550 km), which might lead to the formation of the detached haze layer through
a positive feedback process. The new photochemical model using the retrieved prole are in
better agreement with the Cassini measurements than the previous 1D chemistry-diusion
models. Given the retrieved eddy diusion prole, we perform a linear perturbation test to
identify key reactions that control the abundance of C H in the upper stratosphere (∼200
2
2
km). New expressions of the rate coecients for two three-body reactions are proposed so as
to bring the simulated prole closer to the observation. The revised rate coecients should
be investigated in the laboratory and applied to the hydrocarbon chemistry of giant planets.
Keywords:
∗
Titan, Eddy diusion prole, Chemical kinetics
Corresponding author. Tel.:+1-626-818-1153
Preprint submitted to Elsevier
November 30, 2012
highlights.PDF
•
•
•
•
The complete eddy diffusion profile for Titan is derived for the first time
A low eddy diffusion zone is found in the eddy profile
A positive feedback process is proposed to explain the detached haze layer
New reaction rate coefficient is suggested for photochemical modeling
Li.Yuk.12.titan.submit.PDF
Click here to download (Revised) Manuscript, clean version: Li.Yuk.12.titan.submit.PDF
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Click here to view linked References
Eddy Diusion Prole of Titan's Atmosphere Revealed by Cassini
Observations
Cheng Li
a,
∗
a
b,c
a
, Xi Zhang , Mao-Chang Liang , Yuk L. Yung
a California Institute of Technology, 1200 E. California Blvd, MC 150-21, Pasadena, CA 91125,
b Research Center for Environmental Changes, Academia Sinica, Taipei, Taiwan
c Graduate Institute of Astronomy, National Central University, Zhongli, Taiwan
USA
Abstract
The recent measurements from the limb-view soundings of Cassini/CIRS and the stellar
occultations from Cassini/UVIS revealed the complete vertical proles of minor species (e.g.,
C2 H2 and C2 H4 ) from 100 to 1000 km in the atmosphere of Titan. In this study, we developed
an inversion technique to retrieve the eddy diusion prole using as a tracer species. We nd
that the new eddy prole features a low eddy diusion zone near the altitude of detached
haze layer (∼550 km), which might lead to the formation of the detached haze layer through
a positive feedback process. The new photochemical model using the retrieved prole are in
better agreement with the Cassini measurements than the previous 1D chemistry-diusion
models. Given the retrieved eddy diusion prole, we perform a linear perturbation test to
identify key reactions that control the abundance of C2 H2 in the upper stratosphere (∼200
km). New expressions of the rate coecients for two three-body reactions are proposed so as
to bring the simulated prole closer to the observation. The revised rate coecients should
be investigated in the laboratory and applied to the hydrocarbon chemistry of giant planets.
Keywords:
∗
Titan, Eddy diusion prole, Chemical kinetics
Corresponding author. Tel.:+1-626-818-1153
Email address: cli@gps.caltech.edu (Cheng Li)
Preprint submitted to Elsevier
November 30, 2012
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1. Introduction
Eddy diusivity, a parametrization of large-scale mixing processes, is crucial to onedimensional photochemical modeling. This parameter is either estimated by the amplitude
of gravity waves (Lindzen, 1981) or by tting the measured abundance of species whose
vertical structure is controlled primarily by transport (Allen et al., 1981).
In the earlier
photochemical modeling of Titan's atmosphere (Strobel, 1974; Allen et al., 1980; Yung et al.,
1984), eddy diusivities were chosen to be similar to the Jovian values and they were generally
proportional to the inverse square root of background atmospheric density. However, due to
large thermal variations and wind shear in Titan's atmosphere (Fulchignoni et al., 2005),
calculation of the eddy mixing is very uncertain theoretically.
Voyager (Coustenis et al., 1989) and ground-based millimeter observations (Tanguy et al.,
1990) detected HCN and drove the renement of previous eddy diusivity. HCN was thought
to possess low reactivity with other species and its abundance was used to constraint the eddy
diusivity in the lower atmosphere in the models of Toublanc et al. (1995) and Lara et al.
(1996). Wilson and Atreya (2004) determined the nominal eddy diusion prole by creating
a set of 100 monotonically increasing, randomly generated eddy diusion proles to t CH4
and HCN. However, the diculties Lara et al. (1996) and Vinatier et al. (2007) encountered
in simultaneously matching the proles of HCN and C2 -hydrocarbons implied that HCN
chemistry may have been incomplete in their models. Later, Lavvas et al. (2008a) explored
this approach and proposed HCN loss in the lower atmosphere due to polymerization. However, major uncertainty still remains. As a result, HCN is now ruled out as a proxy for the
estimation of eddy diusion.
The arrival of Cassini spacecraft in 2005 and the subsequent measurements provided
new constraints for the eddy diusion prole. Lavvas et al. (2008a,b) constructed the rst
comprehensive photochemical model based on Cassini measurements.
In this model, the
eddy diusion prole was adjusted to t the abundance of C2 H6 (Vinatier et al., 2007) and
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Ar (Waite Jr et al., 2005). In addition, Yelle et al. (2008) suggested a new asymptotical
expression for the eddy diusion prole based on the thermospheric prole of
1000 km) and the stratospheric abundance of C2 H6 (100
explaining the abundance of
∼ 300 km).
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Ar (above
Because of the success in
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Ar, this kind of eddy diusion proles is widely used in recent
chemical models (Vuitton et al., 2008; Hörst et al., 2008; Krasnopolsky, 2009).
the asymptotical expression relies on a free parameter
γ,
However,
which is 0.9 in Yelle et al. (2008)
but 2.0 in the appendix model of Krasnopolsky (2009), and the modeling results produced
by dierent choice of
γ
were inconsistent with the observations (e.g., C2 H2 in the appendix
model of Krasnopolsky, 2009; C4 H2 in Vuitton et al., 2008). The discrepancies are caused
by the unconstrained eddy diusion prole in the mesosphere of Titan (500
∼ 1000
km).
Changes in the altitude of the fall-o region in the asymptotical expression, determined by
γ,
could have a huge impact on the modeling of hydrocarbons.
Recently, more constraints have been placed on the abundance of hydrocarbons in the
mesosphere of Titan (500
∼ 1000
et al., 2011; Kammer et al., 2011).
km) from Cassini/UVIS stellar occultations (Koskinen
Combined with the updated version of Cassini/CIRS
limb view (Vinatier et al., 2010), this is the rst time that we have the measurements of the
complete prole of hydrocarbons, which oer the unprecedented advantage of retrieving the
whole eddy diusion prole. In this work, C2 H2 is selected as the tracer species. The eddy
diusion prole is retrieved from the abundance of C2 H2 and applied to the photochemical
modeling of other species.
Given the revised eddy diusion prole, we can successfully
match the modeling results with the observations except for C2 H4 . In this case, we have used
a modied expression of the chemical reaction rate constants for two key reactions so as to
minimize the dierence between the simulated prole and the observation.
In the following sections, we start with a brief description of the Cassini/CIRS and
Cassini/UVIS data that provide the constraints for the eddy diusivity. Next, we explain
the reason for selecting C2 H2 as the tracer for retrieval, followed by a presentation of our
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photochemical model and retrieval algorithm. Then, the model results are compared with
observations.
Detailed discussion of the results and the possible dynamical interpretation
of the derived eddy diusion prole are provided. The nal part is the revision of chemical
reaction rate coecients.
2. Eddy diusion prole
2.1. Observations
Vinatier et al. (2010) derived the vertical mixing ratio proles of C2 H2 , C2 H4 , C2 H6 ,
C3 H8 , CH3 C2 H, C4 H2 , C6 H6 , HCN, HC3 N and CO2 from Cassini/CIRS limb data acquired
between February 2005 and May 2008. These data covered 9 latitudes from
10 altitudes from
∼150
to
∼450
56◦ S to 80◦ N and
km. In addition, Koskinen et al. (2011) and Kammer et al.
(2011) retrieved density proles of CH4 , C2 H2 , C2 H4 , C4 H2 , HCN, HC3 N and C6 H6 from
Cassini/UVIS stellar occultation data. These data generally covered latitudes from
45◦ N
and altitudes from
∼450
to
∼1000
35◦ S
to
km. In order to increase the signal to noise ratio
and minimize the random error from a single yby, the average concentrations of all ybys
are used. The combination of the limb and occultation data sets establishes the basis for our
eddy diusion prole retrieval.
2.2. Selection of a tracer species
The vertical distribution of a minor species is aected by the chemistry and the eddy diffusion prole. However due to the lack of sucient laboratory measurements and theoretical
calculations, we were obliged to estimate the rate coecients for several reactions, bringing
large uncertainties to the retrieval of eddy diusivity. The selection of a tracer species should
be based on a minimization of this uncertainty. We dene the sensitivity of the abundance
of a species to reaction rate coecients as:
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S=
2
Nr X
Nv X
∆xi
j=1 i=1
where
Nr
is the total number of reactions;
∆xi
xi
i
when the rate coecient of reaction
Nv
xi
(1)
j
is the total number of vertical levels, and
stands for the fractional change of the abundance for a species,
j
is doubled.
x,
at vertical level
The summation of the square of the
fractional change (S ) over every altitude and reaction gives the sensitivity to reaction rate
coecients. Figure 1 shows the sensitivity for each species.
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Ar and CH4 act as the best
mixing tracers to infer the eddy diusivity around the homopause at about 800 km, where
the eddy diusivity equals the molecular diusivity. Because the abundance of CH4 can be
inuenced either by diusion or by escape (Yelle et al., 2008; Strobel, 2009), we constrain the
eddy diusivity around the homopause by the abundance of
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Ar. In the lower atmosphere,
the chemical destruction timescales for those species are so long that their mixing ratio is
almost constant and is unaected by the value of eddy diusivity. Therefore, besides
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Ar,
C2 H2 is chosen as a tracer species to derive the eddy diusivity in the lower atmosphere as
indicated in Figure 1. Actually, C2 H2 has several merits as a mixing tracer.
First, it is the parent species that drives the whole reaction chain in the stratosphere via
photosensitized dissociation of CH4 (Yung et al., 1984, section IIb). Once its abundance is
determined, it is easy to obtain the correct abundance for the related species (e.g., C2 H4 ,
C2 H6 ). This will be discussed later in detail in section 3.
Second, the prole of C2 H2 in the stratosphere puts a stronger constraint on the eddy
diusivities than C2 H6 . Both C2 H2 and C2 H6 have a constant mixing ratio in the stratosphere,
but the lifetime of C2 H2 (∼
2 × 108 s
) is much smaller than that of C2 H6 (∼
1 × 1010 s
). In
order to maintain the constant mixing ratios, the eddy diusion timescale must be less than
the shorter lifetime. Thus, the eddy diusion prole constrained by C2 H6 could match the
prole of C2 H6 (Lavvas et al., 2008b, Fig. 2), but failed to match the prole of C2 H2 (Lavvas
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et al., 2008b, Fig. 4). In contrast, the eddy diusion prole constrained by C2 H2 is able to
match both proles.
Third, unlike C2 H6 , the spectral feature of C2 H2 does not overlap with CH4 in the FarUltraviolet (FUV) region; therefore it can be detected both by FUV stellar occultation and
CIRS limb view, which in combination reveals a complete prole in the vertical for the
retrieval.
2.3. Photochemical model and retrieval algorithm
The one-dimensional Caltech/JPL photochemical model is used for the modeling of Titan's atmosphere. Chemical reactions in the model are taken from Moses et al. (2005). CH4 is
allowed to escape from the top of atmosphere at the rate of
2.4×109 cm−2 s−1 , consistent with
the value derived by Strobel (2009) and Yelle et al. (2008). Only neutral hydrocarbons and
nitriles are included in the present model since the ion reaction at the top of the atmosphere
can have only a minor eect on the abundance of C2 H2 , and these eects do not propagate
into the lower atmosphere where the retrieval of eddy diusion prole is performed. However,
ion-neutral reactions have a profound impact on the formation of benzene (Vuitton et al.,
2009). In order to compensate this pathway, we added an articial source of benzene from
909 to 1085 km with a production rate of
by this articial source is about
Vuitton et al. (2009).
0.5 cm−3 s−1 .
107 cm−2 s−1 ,
The integral production rate given
which is consistent with the value derived by
The temperature and density proles are based on Huygens/HASI
probe (Fulchignoni et al., 2005). Large thermal waves in the vertical are removed and the
temperature is set to an isothermal value of 170 K from 500 to 1400 km. The model simulation is diurnally averaged at mid-latitudes and the incident UV ux is the mean between
solar maximum and minimum.
For the purpose of retrieving the eddy diusion prole, we choose the Levenberg-Marquardt
algorithm (Moré, 1978), a standard non-linear least square optimization method. It does not
require the a priori for the retrieval and that avoids any possible bias toward the solution.
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The cost function is the square of the dierence between the model and the observations:
J = ||(xn (β) − xnobs )T Sobs (xn (β) − xnobs )||
where
x stands for the log abundance; β
(2)
is the eddy diusion prole and is approximated by a
cubic spline interpolation function sampled at 10 equally spaced levels from 130 to 1000 km.
For altitudes lower than 500 km,
xnobs
is the mean value of 7 non-polar measurements from
Cassini/CIRS limb view (Vinatier et al., 2010); for altitudes above 500 km and below 1000
km,
xnobs
is the mean value of 5 Cassini/UVIS stellar occultation measurements (Kammer
et al., 2011); for altitudes above 1000 km,
et al., 2008).
Sobs
β.
is the measured concentration for
is the observational error covariant matrix.
elements are neglected. The vector
prole
xnobs
xn (β)
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Ar (Yelle
For simplicity, o-diagonal
models the concentration based on eddy diusion
Eddy diusivities at levels higher than 1000 km are set equal to the boundary value
at 1000 km. Eddy diusivities at levels lower than 130 km are set to
5 × 103 cm2 s−1 ,
small
enough to maintain the large abundance of C2 H2 and C2 H6 above the condensation region
and consistent with the value used in Lavvas et al. (2008a) and Krasnopolsky (2009). The
initial guess of the eddy diusion prole is chosen to be an empirical curve of eddy diusion
predicted by the inverse square root of density (Liang et al., 2007). Its value is approximately
5 × 103 cm2 s−1
at the tropopause and
1 × 108 cm2 s−1
at the top of the atmosphere (Figure
3, blue dashed line). The adjustment of eddy diusivity is made according to the LevenbergMarquardt algorithm so as to minimize the cost function.
In order to estimate the error of the retrieval, we used the bootstrap method (Press
et al., 1992), a Monte Carlo technique of error estimation. Suppose the mean value of the
observations is
D0
with
N
data points and the standard deviation is
α = 0.3 so as to cover the spread of dierent ybys.
data sets
D1 , D2 , D3
. . . , also with
N
αD0 .
In our case,
We generate a huge number of synthetic
data points, by randomly perturbing the observation
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(D0 ) at each level following a Gaussian distribution:
to replace the original data set
D0
N (D0 , αD0 ).
The procedure is simply
with the synthetic data sets and do the same retrieval
again, obtaining a set of synthetic eddy diusion proles.
The synthetic data sets can be
viewed as a possible realization of a random variable and the distribution of synthetic eddy
diusion proles provides the condence interval of the retrieved results. In order to have a
statistically signicant result, we created a synthetic set of 1148 cases.
2.4. Retrieval results and comparisons with observations
The retrieved eddy diusivity prole is shown in Figure 3 (blue solid line). It is signicantly dierent from our initial guess. The eddy diusivity is small in the lower stratosphere
but increases sharply by 3 orders of magnitude from 200 to 300 km, implying a very turbulent
atmospheric condition. From 300 to 600 km the eddy diusivity decreases with altitude and
creates a local minimum at around 600 km, implying a stable layer in Titan's atmosphere.
The dierence between the maximum value at 300 km and the minimum value at 600 km
is over 2 times bigger than the largest standard deviation in this interval, which makes it a
robust feature. Above this local minimum, the eddy diusivity increases again with altitude
and reaches
7 × 108 cm2 s−1
in the upper atmosphere. The retrieval algorithm begins to lose
sensitivity at higher altitude because the molecular diusivity dominates in this region (red
line).
(
40
Figure 4 shows the vertical prole of CH4 and the tracer species used for retrieval
Ar, C2 H2 ). Previous modeling results (Lavvas et al., 2008b; Krasnopolsky, 2009) are also
compared alongside (dashed lines).
CH4 escape is required to match the observed abun-
20%)
between the modeling prole and INMS measurements
dance. The dierence (about
(Yelle et al., 2008) is mainly due to large uncertainties and variations of the N2 prole in
the thermosphere.
The mixing ratio of C2 H2 is about 3 ppm from 100 to 500 km and is
captured by our model (red line) and Lavvas et al. (2008b)'s model (blue line).
However
Lavvas et al. (2008b) underestimated the abundance of C2 H2 in the mesosphere due to the
unconstrained eddy prole in this region. Using the new eddy prole constrained by FUV
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stellar occultation, our model provides the best t to the observations.
Figure 5 shows the prole of other hydrocarbons using two eddy diusivity proles: the
old one (blue dashed line in Figure 3) and the retrieved one (blue solid line in Figure 3). The
results are discussed below.
C2 H4 : The mixing prole of ethylene does not change much when the new eddy diusion
prole is used.
This is because ethylene is highly reactive in Titan's atmosphere and its
chemical timescale (∼
(∼
109 s
107 s
at 100 km) is short compared to the eddy diusion timescale
at 100 km). The prole of C2 H4 can be largely explained by local chemical equi-
librium and is insensitive to eddy diusion. The stratospheric mixing ratio of C2 H4 is about
10−8
while the observation from Cassini/CIRS gives a mixing ratio of
10−7 ,
10 times big-
ger than the model. The discrepancy between the simulated prole and observations in the
stratosphere has been known in the previous modeling (e.g., Lebonnois et al., 2001; Wilson
and Atreya, 2004; Lavvas et al., 2008b). Surprisingly, it occurred not only in several photochemical models for Titan but in the photochemical model for giant planets as well (Moses
et al., 2005: Figure 14 for Jupiter, Figure 31 for Saturn and Figure 32 for Neptune; Moses
and Greathouse, 2005: Figure 6). Only Lara et al. (1996)'s model could produce the right
amount of ethylene in the stratosphere by articially imposing an upward ux of C2 H4 from
the ground.
Lavvas et al. (2008b) suggested that enhancement of ethylene was caused by
the downwelling branch of Hadley circulation. Two major problems could be raised with this
scenario. First, Cassini/CIRS observations (Vinatier et al., 2010) show that the latitudinal
concentration gradient of a species is weak outside the polar vortex (∼
45◦ N).
Yet, inside
the polar vortex, where the dynamical mixing is prohibited by the barrier of steep horizontal
potential vorticity gradient (Teanby et al., 2008), the concentration is enhanced by a factor of
5 to 10. Therefore, the enhancement of C2 H4 through Hadley circulation would probably be
conned in the polar region and do not aect tropics and midlatitudes. Second, the mixing
ratio of C2 H4 exhibits a clear decrease with altitude, which could not be explained by the
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Hadley circulation scenario if the production zone of C2 H4 were only in the mesosphere. In
consequence, the negative mixing gradient of C2 H4 in lower latitudes disproves the downwelling scenario and calls for an additional source in the stratosphere. This source can be
either the horizontal advection from the winter pole towards the equator (Crespin et al., 2008)
or a stronger three-body recombination in the stratosphere. Since the problem is universal
in the photochemical modeling of the outer solar systems, we suggest that it occurs because
of the incorrect reaction rate coecients for three-body reactions.
tions of the rate coecients for two key reactions (R109) H
(R113) H
+C
2
H4
+ M −→ C
2
H5
+ M.
+C
2
H2
We proposed modica-
+ M −→ C
2
H3
+M
and
The modeling result (red line) matches the observed
stratospheric abundance. Detailed discussion of the modications is provided in section 3.
C2 H6 : Ethane is a long-lived species.
It is hard to break it apart once formed.
As in
the case of C2 H2 , the large eddy diusivity retrieved in the lower atmosphere brings the
abundance of ethane close to the observed values. Unfortunately, the overlapping absorption
cross section of ethane with that of methane makes it hard to determine its abundance in the
upper atmosphere from FUV stellar occultation. Cassini/INMS gives a puzzling low mixing
−5
ratio (4.6 × 10
from Magee et al., 2009 in contrast to
3 × 10−4
in the model), which might
be the result of other processes (e.g., ion-related reactions), not included in the model, being
responsible for the destruction of C2 H6 in the upper atmosphere.
CH3 C2 H and C3 H8 : The mixing ratios of methylacetylene and propane agree with the
Cassini/ CIRS measurements in the lower atmosphere, but are dierent from the Cassni/INMS
result in the upper atmosphere. The model overestimates the abundance of CH3 C2 H by a
factor of two and underestimates the abundance of C3 H8 by one order of magnitude. The disagreement with the INMS results was rst noticed by Lavvas et al. (2008b), yet the possible
reasons for the underestimation are still unknown.
C4 H2 and C6 H6 : Modeling results of diacetylene and benzene match very well with the
observations when the new eddy diusion prole is used.
10
At around 400 km, our model
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overestimated C4 H2 . This is probably due to the lack of the haze formation process as a sink
for C4 H2 (Lavvas et al., 2008b). The prole of benzene is not sensitive to eddy transport.
With the aid of an articial source around 1000 km imitating the ion reactions, our models
are able to reproduce the abundance of benzene observed by Cassni/INMS and FUV stellar
occultation.
In general, large eddy diusivity in the lower atmosphere is essential to force the mixing
ratio of a species to remain constant despite the large dierence in chemistry; the existence
of a stable layer (low eddy diusion zone) in the middle of the atmosphere enables chemical production to take place and keeps the simulated prole close to the observations.
A
photochemical model using the new eddy diusivity agrees better with observations.
A possible dynamical interpretation of the eddy diusion prole is based on the eect of
aerosol heating. Liang et al. (2007) rst observed tholins that are probably the intermediates
in the formation of aerosols at 1000 km from UVIS. These particles, if produced by ion reaction at top of the atmosphere, could be transported downward quickly and grow slowly when
the eddy diusivity is large. However, as the particle density increases, they could absorb
enough solar heat to create a local temperature inversion. Lavvas et al. (2009) estimated a
heating rate of aerosols which results in a temperature inversion of about 20 K (Figure 2, red
lines). The temperature inversion layer stabilizes the atmosphere, creating a low eddy diusion zone under it, and slows the downward transport of aerosols. The aerosols are retained
in this layer and particles could grow more rapidly through fractal aggregation (Lavvas et al.,
2009), absorbing more heat as a result. This kind of positive feedback mechanism predicts
that the vertical mixing intensity would decrease in the inversion layer (at about 500 km),
which is manifested by a decrease in the eddy diusivity (Figure 2, blue line). This fractal
growth of aerosol permitted by a slow vertical transport in a stable layer might explain the
existence of the detached haze layer observed by Cassini/UVIS.
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3. Chemical reaction rate coecients
Reaction rate coecients for hydrocarbon chemistry at low pressure (< 1 hpa) and low
temperature (< 200 K) are often not available from laboratory measurements.
We used
a pathway analysis and linear perturbation method to determine the key reactions that
control the abundance of C2 H4 in the stratosphere. Figure 6 shows the important reactions
for the production and loss of C2 H4 at each level.
In the thermosphere and mesosphere,
C2 H4 forms mainly through the insertion of the CH radical into CH4 .
When the photons
are used up by methane photolysis and atmospheric density increases enough, three-body
reaction H
+C
2
H3
+ M −→ C
2
H4
+M
is initiated and gradually takes over the production
of C2 H4 . A large portion of C2 H4 loss is photodissociation into C2 H2 through the reaction
C2 H4
−→ C
2
H2
+ H /2 H.
2
However, in the lower atmosphere, where the atmospheric density
is high and high energy photons are scarce, loss due to three-body reaction H
M
−→ C
2
H5
+M
+C
2
H4
+
becomes more ecient and accounts for almost 80% to the total loss rate
around 200 km. The same analysis has been done for other C2 -hydrocarbons and the result
is illustrated in Figure 7. In short, C2 H2 , transported downward from the production region,
is the parent of all species in the lower atmosphere where it combines with H atoms to
gradually form C2 H3 , C2 H4 and C2 H5 .
Therefore, the atomic H combination chain is the
key to determine the abundance of C2 H4 in the lower atmosphere.
Figure 8 shows the
fractional change of C2 H4 when each reaction rate coecient is doubled. It is clear that the
abundances of C2 H4 in the lower atmosphere are most sensitive to two three-body reactions:
(R109) H + C2 H2
+ M −→ C
2
H3
+M
and (R113) H + C2 H4
+ M −→ C
2
H5
+ M.
Moses et al.
(2000) and Moses et al. (2005) use dierent rate coecients for R109. The expression used in
Moses et al. (2000) matches better with the laboratory measurement (Figure 9). Therefore
we use the Moses et al. (2000)'s version of rate coecients for R109. R131 used in Moses et al.
(2005) model do not agree with the laboratory measurement at higher temperatures (Figure
10). We propose a new expression for reaction rate coecients for this reaction to match the
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observations at all measured temperatures (Figure 8, Table 1). The new expression for the
rate coecient requires a lower reaction rate coecient at Titan's temperature.
4. Conclusion
In this work, we retrieved the eddy diusion prole based on the vertical distribution
of C2 H2 and the abundance of
diusion zone at 600 km.
40
Ar.
The new eddy diusion prole features a low eddy
This could be a result of aerosol heating and could provide a
positive feedback for the formation of the detached haze layer. Similar low eddy diusion
zones were also found in Earth's atmosphere (Allen et al., 1981) due to the gravity wave and
tidal breakdown (Lindzen, 1981).
Therefore, the common assumption of a monotonically
increasing eddy diusion prole may not be valid for terrestrial planets. Figure 11 provides
a comparison with the eddy diusion proles used in previous models. For practical use, we
suggest a fourth-order polynomial expression for the eddy diusion prole:
Ke (z) =
where
Ke



5 × 103
z<1


e−5.22+15.7z−3.97z2 +0.401z3 −0.0137z4
1 6 z < 10
(3)
is the eddy diusivity in units of
cm2 s−1 ; z
is the altitude in units of 100 km.
Chemical kinetics are examined and modied so as to match the observations of C2 H4 in
the stratosphere. The new expression agrees with the laboratory measurements at all high
temperatures and pressures. This might help to solve a similar problem for the photochemical modeling of C2 H4 on giant planets.
We suggest new laboratory measurements at low
temperature and low pressure to settle this issue.
Acknowledgements
This research was supported in part by the Cassini UVIS program via NASA grant
JPL.1459109 to the California Institute of Technology.
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part by NASA NNX09AB72G grant to the California Institute of Technology.
We thank
Michael Line for the discussion on the retrieval algorithm and the bootstrap method for
error analysis.
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Table 1: Summary of reaction rate coecients. M represents any third body. Two-body rate constants and
high-pressure limiting rate constants for three-body reactions (k∞ ) are in units of cm3 s−1 . Low-pressure
limiting rate constants for three-body reactions (k0 ) are in units of cm6 s−1 .
Table 2: Photochemical reaction list. See Moses et al. (2005) for reaction rate coecients and references
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Figure 1: Summation of the square of fractional changes over 82 levels and 297 reactions for 18 species when
the rate coecient for each reaction is doubled. For clarity, radicals are not shown in the gure because their
abundances are always not aected by transport due to small chemical destruction timescales. The total
sensitivity is divided into three parts: red part is the contribution from 50 km to 500 km; magenta, from 500
km to 1000 km; yellow, above 1000 km.
Figure 2: Retrieved eddy diusion prole (blue line) plotted with the eect of aerosol heating. Red dashed
line is the temperature prole without aerosol heating; red solid line is the temperature prole with the
aerosol heating by a detached hazed layer (Lavvas et al., 2009).
Figure 3: The eddy diusion prole retrieved from the abundances of C2 H2 and 40Ar (blue sold line) along
with the molecular diusion coecients for CH4 (red line) and the initial guess of the eddy diusion prole
(blue dashed line). Solid black line is the median value of the synthetic set of eddy diusion proles. Shaded
region shows the 1-σ condence level around the median.
Figure 4: Model calculated vertical prole of the species ( 40Ar, C2 H2 ) used for the retrieval of the eddy
diusion prole along with the CH4 prole. Magenta points with error bars are the INMS measurements
(Yelle et al., 2008; Magee et al., 2009). Black dots are the measurements from FUV stellar occultation.
Dashed lines are the modeled proles for C2 H2 from Lavvas et al., 2008b (blue line) and Krasnopolsky, 2009
(basic model: green line, appendix model: cyan line). Red line is our model using the retrieved eddy diusion
prole.
Figure 5: The mixing ratios of the major hydrocarbons in dierent models. Blue dashed line is the result
using empirical eddy diusion prole (Liang et al., 2007); blue solid line is the result using the retrieved
eddy diusion prole; red line is the result using the retrieved eddy diusion prole and with the revised
rate coecients for R109 and R114. The observations are plotted along with the model results. Dark dots
in the upper atmosphere are from FUV stellar occultation (Kammer et al., 2011). Dark dots in the lower
atmosphere are from CIRS limb view (Vinatier et al., 2010). Magenta points at about 1000 km are from
INMS (Magee et al., 2009).
Figure 6: Production and loss rates of C2 H4 plotted in percentage of the total rate. Only the three most
important reactions are plotted for clarity. Dierent reactions are identied by dierent colors according
to the labels in the gure. At each level, the area of a color patch equals to its contribution to the total
production or loss rate.
Figure 7: Main chemical scheme for C2 -hydrocarbons. Yellow arrow indicates photodissociation is involved.
Red arrow indicates three-body reaction is involved. Other reactions are shown in blue.
Figure 8: Color shows the fractional change of C2 H4 when each reaction rate coecient is doubled. The
abscissa is the reaction number in the model. Any reaction with a number larger than 120 is not important
and is omitted for clarity. The 120 reactions are listed in Table ??
Figure 9: Rate coecients for the reaction H + C2 H2 + M −−→ C2 H3 + M. Colored dots with error bars are the
laboratory measurements from Payne and Stief (1976). Temperatures indicated by the color are listed in the
gure. Dashed lines are the reaction rate coecients used in Moses et al. (2005); solid lines are the reaction
rate coecients used in Moses et al. (2000). No laboratory measurements are available for temperatures on
Titan (∼170 K) and in the pressure range of the stratosphere (∼0.1 hpa).
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Figure 10: Rate coecients for the reaction H + C2 H4 + M −−→ C2 H5 + M. Colored dots with error bars
are the laboratory measurements from Lightfoot and Pilling (1987). Temperatures indicated by the color are
listed in the gure. Dashed lines are the reaction rate coecients used in Moses et al. (2005); solid lines are
the new reaction rate coecients (see Table 1). No laboratory measurements are available for temperatures
on Titan (∼170 K) and in the pressure range of the stratosphere (∼0.1 hpa).
Figure 11: Comparison of dierent eddy diusion proles used in previous photochemical modelings: red
(Yelle et al., 2008); green (Lavvas et al., 2008b); blue (Lara et al., 1996); cyan (this work); Dashed line
(molecular diusivity for CH4 ).
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Li.Yuk.12.titan.submit.tex
Click here to download (Revised) Manuscript, clean version: Li.Yuk.12.titan.submit.tex
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Click here to view linked References
Eddy Diusion Prole of Titan's Atmosphere Revealed by Cassini
Observations
Cheng Li
a,
∗
a
b,c
a
, Xi Zhang , Mao-Chang Liang , Yuk L. Yung
a California Institute of Technology, 1200 E. California Blvd, MC 150-21, Pasadena, CA 91125,
b Research Center for Environmental Changes, Academia Sinica, Taipei, Taiwan
c Graduate Institute of Astronomy, National Central University, Zhongli, Taiwan
USA
Abstract
The recent measurements from the limb-view soundings of Cassini/CIRS and the stellar
occultations from Cassini/UVIS revealed the complete vertical proles of minor species (e.g.,
C2 H2 and C2 H4 ) from 100 to 1000 km in the atmosphere of Titan. In this study, we developed
an inversion technique to retrieve the eddy diusion prole using as a tracer species. We nd
that the new eddy prole features a low eddy diusion zone near the altitude of detached
haze layer (∼550 km), which might lead to the formation of the detached haze layer through
a positive feedback process. The new photochemical model using the retrieved prole are in
better agreement with the Cassini measurements than the previous 1D chemistry-diusion
models. Given the retrieved eddy diusion prole, we perform a linear perturbation test to
identify key reactions that control the abundance of C2 H2 in the upper stratosphere (∼200
km). New expressions of the rate coecients for two three-body reactions are proposed so as
to bring the simulated prole closer to the observation. The revised rate coecients should
be investigated in the laboratory and applied to the hydrocarbon chemistry of giant planets.
Keywords:
∗
Titan, Eddy diusion prole, Chemical kinetics
Corresponding author. Tel.:+1-626-818-1153
Email address: cli@gps.caltech.edu (Cheng Li)
Preprint submitted to Elsevier
November 30, 2012
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1. Introduction
Eddy diusivity, a parametrization of large-scale mixing processes, is crucial to onedimensional photochemical modeling. This parameter is either estimated by the amplitude
of gravity waves (Lindzen, 1981) or by tting the measured abundance of species whose
vertical structure is controlled primarily by transport (Allen et al., 1981).
In the earlier
photochemical modeling of Titan's atmosphere (Strobel, 1974; Allen et al., 1980; Yung et al.,
1984), eddy diusivities were chosen to be similar to the Jovian values and they were generally
proportional to the inverse square root of background atmospheric density. However, due to
large thermal variations and wind shear in Titan's atmosphere (Fulchignoni et al., 2005),
calculation of the eddy mixing is very uncertain theoretically.
Voyager (Coustenis et al., 1989) and ground-based millimeter observations (Tanguy et al.,
1990) detected HCN and drove the renement of previous eddy diusivity. HCN was thought
to possess low reactivity with other species and its abundance was used to constraint the eddy
diusivity in the lower atmosphere in the models of Toublanc et al. (1995) and Lara et al.
(1996). Wilson and Atreya (2004) determined the nominal eddy diusion prole by creating
a set of 100 monotonically increasing, randomly generated eddy diusion proles to t CH4
and HCN. However, the diculties Lara et al. (1996) and Vinatier et al. (2007) encountered
in simultaneously matching the proles of HCN and C2 -hydrocarbons implied that HCN
chemistry may have been incomplete in their models. Later, Lavvas et al. (2008a) explored
this approach and proposed HCN loss in the lower atmosphere due to polymerization. However, major uncertainty still remains. As a result, HCN is now ruled out as a proxy for the
estimation of eddy diusion.
The arrival of Cassini spacecraft in 2005 and the subsequent measurements provided
new constraints for the eddy diusion prole. Lavvas et al. (2008a,b) constructed the rst
comprehensive photochemical model based on Cassini measurements.
In this model, the
eddy diusion prole was adjusted to t the abundance of C2 H6 (Vinatier et al., 2007) and
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Ar (Waite Jr et al., 2005). In addition, Yelle et al. (2008) suggested a new asymptotical
expression for the eddy diusion prole based on the thermospheric prole of
1000 km) and the stratospheric abundance of C2 H6 (100
explaining the abundance of
∼ 300 km).
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Ar (above
Because of the success in
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Ar, this kind of eddy diusion proles is widely used in recent
chemical models (Vuitton et al., 2008; Hörst et al., 2008; Krasnopolsky, 2009).
the asymptotical expression relies on a free parameter
γ,
However,
which is 0.9 in Yelle et al. (2008)
but 2.0 in the appendix model of Krasnopolsky (2009), and the modeling results produced
by dierent choice of
γ
were inconsistent with the observations (e.g., C2 H2 in the appendix
model of Krasnopolsky, 2009; C4 H2 in Vuitton et al., 2008). The discrepancies are caused
by the unconstrained eddy diusion prole in the mesosphere of Titan (500
∼ 1000
km).
Changes in the altitude of the fall-o region in the asymptotical expression, determined by
γ,
could have a huge impact on the modeling of hydrocarbons.
Recently, more constraints have been placed on the abundance of hydrocarbons in the
mesosphere of Titan (500
∼ 1000
et al., 2011; Kammer et al., 2011).
km) from Cassini/UVIS stellar occultations (Koskinen
Combined with the updated version of Cassini/CIRS
limb view (Vinatier et al., 2010), this is the rst time that we have the measurements of the
complete prole of hydrocarbons, which oer the unprecedented advantage of retrieving the
whole eddy diusion prole. In this work, C2 H2 is selected as the tracer species. The eddy
diusion prole is retrieved from the abundance of C2 H2 and applied to the photochemical
modeling of other species.
Given the revised eddy diusion prole, we can successfully
match the modeling results with the observations except for C2 H4 . In this case, we have used
a modied expression of the chemical reaction rate constants for two key reactions so as to
minimize the dierence between the simulated prole and the observation.
In the following sections, we start with a brief description of the Cassini/CIRS and
Cassini/UVIS data that provide the constraints for the eddy diusivity. Next, we explain
the reason for selecting C2 H2 as the tracer for retrieval, followed by a presentation of our
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photochemical model and retrieval algorithm. Then, the model results are compared with
observations.
Detailed discussion of the results and the possible dynamical interpretation
of the derived eddy diusion prole are provided. The nal part is the revision of chemical
reaction rate coecients.
2. Eddy diusion prole
2.1. Observations
Vinatier et al. (2010) derived the vertical mixing ratio proles of C2 H2 , C2 H4 , C2 H6 ,
C3 H8 , CH3 C2 H, C4 H2 , C6 H6 , HCN, HC3 N and CO2 from Cassini/CIRS limb data acquired
between February 2005 and May 2008. These data covered 9 latitudes from
10 altitudes from
∼150
to
∼450
56◦ S to 80◦ N and
km. In addition, Koskinen et al. (2011) and Kammer et al.
(2011) retrieved density proles of CH4 , C2 H2 , C2 H4 , C4 H2 , HCN, HC3 N and C6 H6 from
Cassini/UVIS stellar occultation data. These data generally covered latitudes from
45◦ N
and altitudes from
∼450
to
∼1000
35◦ S
to
km. In order to increase the signal to noise ratio
and minimize the random error from a single yby, the average concentrations of all ybys
are used. The combination of the limb and occultation data sets establishes the basis for our
eddy diusion prole retrieval.
2.2. Selection of a tracer species
The vertical distribution of a minor species is aected by the chemistry and the eddy diffusion prole. However due to the lack of sucient laboratory measurements and theoretical
calculations, we were obliged to estimate the rate coecients for several reactions, bringing
large uncertainties to the retrieval of eddy diusivity. The selection of a tracer species should
be based on a minimization of this uncertainty. We dene the sensitivity of the abundance
of a species to reaction rate coecients as:
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S=
2
Nr X
Nv X
∆xi
j=1 i=1
where
Nr
is the total number of reactions;
∆xi
xi
i
when the rate coecient of reaction
Nv
xi
(1)
j
is the total number of vertical levels, and
stands for the fractional change of the abundance for a species,
j
is doubled.
x,
at vertical level
The summation of the square of the
fractional change (S ) over every altitude and reaction gives the sensitivity to reaction rate
coecients. Figure 1 shows the sensitivity for each species.
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Ar and CH4 act as the best
mixing tracers to infer the eddy diusivity around the homopause at about 800 km, where
the eddy diusivity equals the molecular diusivity. Because the abundance of CH4 can be
inuenced either by diusion or by escape (Yelle et al., 2008; Strobel, 2009), we constrain the
eddy diusivity around the homopause by the abundance of
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Ar. In the lower atmosphere,
the chemical destruction timescales for those species are so long that their mixing ratio is
almost constant and is unaected by the value of eddy diusivity. Therefore, besides
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Ar,
C2 H2 is chosen as a tracer species to derive the eddy diusivity in the lower atmosphere as
indicated in Figure 1. Actually, C2 H2 has several merits as a mixing tracer.
First, it is the parent species that drives the whole reaction chain in the stratosphere via
photosensitized dissociation of CH4 (Yung et al., 1984, section IIb). Once its abundance is
determined, it is easy to obtain the correct abundance for the related species (e.g., C2 H4 ,
C2 H6 ). This will be discussed later in detail in section 3.
Second, the prole of C2 H2 in the stratosphere puts a stronger constraint on the eddy
diusivities than C2 H6 . Both C2 H2 and C2 H6 have a constant mixing ratio in the stratosphere,
but the lifetime of C2 H2 (∼
2 × 108 s
) is much smaller than that of C2 H6 (∼
1 × 1010 s
). In
order to maintain the constant mixing ratios, the eddy diusion timescale must be less than
the shorter lifetime. Thus, the eddy diusion prole constrained by C2 H6 could match the
prole of C2 H6 (Lavvas et al., 2008b, Fig. 2), but failed to match the prole of C2 H2 (Lavvas
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et al., 2008b, Fig. 4). In contrast, the eddy diusion prole constrained by C2 H2 is able to
match both proles.
Third, unlike C2 H6 , the spectral feature of C2 H2 does not overlap with CH4 in the FarUltraviolet (FUV) region; therefore it can be detected both by FUV stellar occultation and
CIRS limb view, which in combination reveals a complete prole in the vertical for the
retrieval.
2.3. Photochemical model and retrieval algorithm
The one-dimensional Caltech/JPL photochemical model is used for the modeling of Titan's atmosphere. Chemical reactions in the model are taken from Moses et al. (2005). CH4 is
allowed to escape from the top of atmosphere at the rate of
2.4×109 cm−2 s−1 , consistent with
the value derived by Strobel (2009) and Yelle et al. (2008). Only neutral hydrocarbons and
nitriles are included in the present model since the ion reaction at the top of the atmosphere
can have only a minor eect on the abundance of C2 H2 , and these eects do not propagate
into the lower atmosphere where the retrieval of eddy diusion prole is performed. However,
ion-neutral reactions have a profound impact on the formation of benzene (Vuitton et al.,
2009). In order to compensate this pathway, we added an articial source of benzene from
909 to 1085 km with a production rate of
by this articial source is about
Vuitton et al. (2009).
0.5 cm−3 s−1 .
107 cm−2 s−1 ,
The integral production rate given
which is consistent with the value derived by
The temperature and density proles are based on Huygens/HASI
probe (Fulchignoni et al., 2005). Large thermal waves in the vertical are removed and the
temperature is set to an isothermal value of 170 K from 500 to 1400 km. The model simulation is diurnally averaged at mid-latitudes and the incident UV ux is the mean between
solar maximum and minimum.
For the purpose of retrieving the eddy diusion prole, we choose the Levenberg-Marquardt
algorithm (Moré, 1978), a standard non-linear least square optimization method. It does not
require the a priori for the retrieval and that avoids any possible bias toward the solution.
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The cost function is the square of the dierence between the model and the observations:
J = ||(xn (β) − xnobs )T Sobs (xn (β) − xnobs )||
where
x stands for the log abundance; β
(2)
is the eddy diusion prole and is approximated by a
cubic spline interpolation function sampled at 10 equally spaced levels from 130 to 1000 km.
For altitudes lower than 500 km,
xnobs
is the mean value of 7 non-polar measurements from
Cassini/CIRS limb view (Vinatier et al., 2010); for altitudes above 500 km and below 1000
km,
xnobs
is the mean value of 5 Cassini/UVIS stellar occultation measurements (Kammer
et al., 2011); for altitudes above 1000 km,
et al., 2008).
Sobs
β.
is the measured concentration for
is the observational error covariant matrix.
elements are neglected. The vector
prole
xnobs
xn (β)
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Ar (Yelle
For simplicity, o-diagonal
models the concentration based on eddy diusion
Eddy diusivities at levels higher than 1000 km are set equal to the boundary value
at 1000 km. Eddy diusivities at levels lower than 130 km are set to
5 × 103 cm2 s−1 ,
small
enough to maintain the large abundance of C2 H2 and C2 H6 above the condensation region
and consistent with the value used in Lavvas et al. (2008a) and Krasnopolsky (2009). The
initial guess of the eddy diusion prole is chosen to be an empirical curve of eddy diusion
predicted by the inverse square root of density (Liang et al., 2007). Its value is approximately
5 × 103 cm2 s−1
at the tropopause and
1 × 108 cm2 s−1
at the top of the atmosphere (Figure
3, blue dashed line). The adjustment of eddy diusivity is made according to the LevenbergMarquardt algorithm so as to minimize the cost function.
In order to estimate the error of the retrieval, we used the bootstrap method (Press
et al., 1992), a Monte Carlo technique of error estimation. Suppose the mean value of the
observations is
D0
with
N
data points and the standard deviation is
α = 0.3 so as to cover the spread of dierent ybys.
data sets
D1 , D2 , D3
. . . , also with
N
αD0 .
In our case,
We generate a huge number of synthetic
data points, by randomly perturbing the observation
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(D0 ) at each level following a Gaussian distribution:
to replace the original data set
D0
N (D0 , αD0 ).
The procedure is simply
with the synthetic data sets and do the same retrieval
again, obtaining a set of synthetic eddy diusion proles.
The synthetic data sets can be
viewed as a possible realization of a random variable and the distribution of synthetic eddy
diusion proles provides the condence interval of the retrieved results. In order to have a
statistically signicant result, we created a synthetic set of 1148 cases.
2.4. Retrieval results and comparisons with observations
The retrieved eddy diusivity prole is shown in Figure 3 (blue solid line). It is signicantly dierent from our initial guess. The eddy diusivity is small in the lower stratosphere
but increases sharply by 3 orders of magnitude from 200 to 300 km, implying a very turbulent
atmospheric condition. From 300 to 600 km the eddy diusivity decreases with altitude and
creates a local minimum at around 600 km, implying a stable layer in Titan's atmosphere.
The dierence between the maximum value at 300 km and the minimum value at 600 km
is over 2 times bigger than the largest standard deviation in this interval, which makes it a
robust feature. Above this local minimum, the eddy diusivity increases again with altitude
and reaches
7 × 108 cm2 s−1
in the upper atmosphere. The retrieval algorithm begins to lose
sensitivity at higher altitude because the molecular diusivity dominates in this region (red
line).
(
40
Figure 4 shows the vertical prole of CH4 and the tracer species used for retrieval
Ar, C2 H2 ). Previous modeling results (Lavvas et al., 2008b; Krasnopolsky, 2009) are also
compared alongside (dashed lines).
CH4 escape is required to match the observed abun-
20%)
between the modeling prole and INMS measurements
dance. The dierence (about
(Yelle et al., 2008) is mainly due to large uncertainties and variations of the N2 prole in
the thermosphere.
The mixing ratio of C2 H2 is about 3 ppm from 100 to 500 km and is
captured by our model (red line) and Lavvas et al. (2008b)'s model (blue line).
However
Lavvas et al. (2008b) underestimated the abundance of C2 H2 in the mesosphere due to the
unconstrained eddy prole in this region. Using the new eddy prole constrained by FUV
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stellar occultation, our model provides the best t to the observations.
Figure 5 shows the prole of other hydrocarbons using two eddy diusivity proles: the
old one (blue dashed line in Figure 3) and the retrieved one (blue solid line in Figure 3). The
results are discussed below.
C2 H4 : The mixing prole of ethylene does not change much when the new eddy diusion
prole is used.
This is because ethylene is highly reactive in Titan's atmosphere and its
chemical timescale (∼
(∼
109 s
107 s
at 100 km) is short compared to the eddy diusion timescale
at 100 km). The prole of C2 H4 can be largely explained by local chemical equi-
librium and is insensitive to eddy diusion. The stratospheric mixing ratio of C2 H4 is about
10−8
while the observation from Cassini/CIRS gives a mixing ratio of
10−7 ,
10 times big-
ger than the model. The discrepancy between the simulated prole and observations in the
stratosphere has been known in the previous modeling (e.g., Lebonnois et al., 2001; Wilson
and Atreya, 2004; Lavvas et al., 2008b). Surprisingly, it occurred not only in several photochemical models for Titan but in the photochemical model for giant planets as well (Moses
et al., 2005: Figure 14 for Jupiter, Figure 31 for Saturn and Figure 32 for Neptune; Moses
and Greathouse, 2005: Figure 6). Only Lara et al. (1996)'s model could produce the right
amount of ethylene in the stratosphere by articially imposing an upward ux of C2 H4 from
the ground.
Lavvas et al. (2008b) suggested that enhancement of ethylene was caused by
the downwelling branch of Hadley circulation. Two major problems could be raised with this
scenario. First, Cassini/CIRS observations (Vinatier et al., 2010) show that the latitudinal
concentration gradient of a species is weak outside the polar vortex (∼
45◦ N).
Yet, inside
the polar vortex, where the dynamical mixing is prohibited by the barrier of steep horizontal
potential vorticity gradient (Teanby et al., 2008), the concentration is enhanced by a factor of
5 to 10. Therefore, the enhancement of C2 H4 through Hadley circulation would probably be
conned in the polar region and do not aect tropics and midlatitudes. Second, the mixing
ratio of C2 H4 exhibits a clear decrease with altitude, which could not be explained by the
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Hadley circulation scenario if the production zone of C2 H4 were only in the mesosphere. In
consequence, the negative mixing gradient of C2 H4 in lower latitudes disproves the downwelling scenario and calls for an additional source in the stratosphere. This source can be
either the horizontal advection from the winter pole towards the equator (Crespin et al., 2008)
or a stronger three-body recombination in the stratosphere. Since the problem is universal
in the photochemical modeling of the outer solar systems, we suggest that it occurs because
of the incorrect reaction rate coecients for three-body reactions.
tions of the rate coecients for two key reactions (R109) H
(R113) H
+C
2
H4
+ M −→ C
2
H5
+ M.
+C
2
H2
We proposed modica-
+ M −→ C
2
H3
+M
and
The modeling result (red line) matches the observed
stratospheric abundance. Detailed discussion of the modications is provided in section 3.
C2 H6 : Ethane is a long-lived species.
It is hard to break it apart once formed.
As in
the case of C2 H2 , the large eddy diusivity retrieved in the lower atmosphere brings the
abundance of ethane close to the observed values. Unfortunately, the overlapping absorption
cross section of ethane with that of methane makes it hard to determine its abundance in the
upper atmosphere from FUV stellar occultation. Cassini/INMS gives a puzzling low mixing
−5
ratio (4.6 × 10
from Magee et al., 2009 in contrast to
3 × 10−4
in the model), which might
be the result of other processes (e.g., ion-related reactions), not included in the model, being
responsible for the destruction of C2 H6 in the upper atmosphere.
CH3 C2 H and C3 H8 : The mixing ratios of methylacetylene and propane agree with the
Cassini/ CIRS measurements in the lower atmosphere, but are dierent from the Cassni/INMS
result in the upper atmosphere. The model overestimates the abundance of CH3 C2 H by a
factor of two and underestimates the abundance of C3 H8 by one order of magnitude. The disagreement with the INMS results was rst noticed by Lavvas et al. (2008b), yet the possible
reasons for the underestimation are still unknown.
C4 H2 and C6 H6 : Modeling results of diacetylene and benzene match very well with the
observations when the new eddy diusion prole is used.
10
At around 400 km, our model
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overestimated C4 H2 . This is probably due to the lack of the haze formation process as a sink
for C4 H2 (Lavvas et al., 2008b). The prole of benzene is not sensitive to eddy transport.
With the aid of an articial source around 1000 km imitating the ion reactions, our models
are able to reproduce the abundance of benzene observed by Cassni/INMS and FUV stellar
occultation.
In general, large eddy diusivity in the lower atmosphere is essential to force the mixing
ratio of a species to remain constant despite the large dierence in chemistry; the existence
of a stable layer (low eddy diusion zone) in the middle of the atmosphere enables chemical production to take place and keeps the simulated prole close to the observations.
A
photochemical model using the new eddy diusivity agrees better with observations.
A possible dynamical interpretation of the eddy diusion prole is based on the eect of
aerosol heating. Liang et al. (2007) rst observed tholins that are probably the intermediates
in the formation of aerosols at 1000 km from UVIS. These particles, if produced by ion reaction at top of the atmosphere, could be transported downward quickly and grow slowly when
the eddy diusivity is large. However, as the particle density increases, they could absorb
enough solar heat to create a local temperature inversion. Lavvas et al. (2009) estimated a
heating rate of aerosols which results in a temperature inversion of about 20 K (Figure 2, red
lines). The temperature inversion layer stabilizes the atmosphere, creating a low eddy diusion zone under it, and slows the downward transport of aerosols. The aerosols are retained
in this layer and particles could grow more rapidly through fractal aggregation (Lavvas et al.,
2009), absorbing more heat as a result. This kind of positive feedback mechanism predicts
that the vertical mixing intensity would decrease in the inversion layer (at about 500 km),
which is manifested by a decrease in the eddy diusivity (Figure 2, blue line). This fractal
growth of aerosol permitted by a slow vertical transport in a stable layer might explain the
existence of the detached haze layer observed by Cassini/UVIS.
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3. Chemical reaction rate coecients
Reaction rate coecients for hydrocarbon chemistry at low pressure (< 1 hpa) and low
temperature (< 200 K) are often not available from laboratory measurements.
We used
a pathway analysis and linear perturbation method to determine the key reactions that
control the abundance of C2 H4 in the stratosphere. Figure 6 shows the important reactions
for the production and loss of C2 H4 at each level.
In the thermosphere and mesosphere,
C2 H4 forms mainly through the insertion of the CH radical into CH4 .
When the photons
are used up by methane photolysis and atmospheric density increases enough, three-body
reaction H
+C
2
H3
+ M −→ C
2
H4
+M
is initiated and gradually takes over the production
of C2 H4 . A large portion of C2 H4 loss is photodissociation into C2 H2 through the reaction
C2 H4
−→ C
2
H2
+ H /2 H.
2
However, in the lower atmosphere, where the atmospheric density
is high and high energy photons are scarce, loss due to three-body reaction H
M
−→ C
2
H5
+M
+C
2
H4
+
becomes more ecient and accounts for almost 80% to the total loss rate
around 200 km. The same analysis has been done for other C2 -hydrocarbons and the result
is illustrated in Figure 7. In short, C2 H2 , transported downward from the production region,
is the parent of all species in the lower atmosphere where it combines with H atoms to
gradually form C2 H3 , C2 H4 and C2 H5 .
Therefore, the atomic H combination chain is the
key to determine the abundance of C2 H4 in the lower atmosphere.
Figure 8 shows the
fractional change of C2 H4 when each reaction rate coecient is doubled. It is clear that the
abundances of C2 H4 in the lower atmosphere are most sensitive to two three-body reactions:
(R109) H + C2 H2
+ M −→ C
2
H3
+M
and (R113) H + C2 H4
+ M −→ C
2
H5
+ M.
Moses et al.
(2000) and Moses et al. (2005) use dierent rate coecients for R109. The expression used in
Moses et al. (2000) matches better with the laboratory measurement (Figure 9). Therefore
we use the Moses et al. (2000)'s version of rate coecients for R109. R131 used in Moses et al.
(2005) model do not agree with the laboratory measurement at higher temperatures (Figure
10). We propose a new expression for reaction rate coecients for this reaction to match the
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observations at all measured temperatures (Figure 8, Table 1). The new expression for the
rate coecient requires a lower reaction rate coecient at Titan's temperature.
4. Conclusion
In this work, we retrieved the eddy diusion prole based on the vertical distribution
of C2 H2 and the abundance of
diusion zone at 600 km.
40
Ar.
The new eddy diusion prole features a low eddy
This could be a result of aerosol heating and could provide a
positive feedback for the formation of the detached haze layer. Similar low eddy diusion
zones were also found in Earth's atmosphere (Allen et al., 1981) due to the gravity wave and
tidal breakdown (Lindzen, 1981).
Therefore, the common assumption of a monotonically
increasing eddy diusion prole may not be valid for terrestrial planets. Figure 11 provides
a comparison with the eddy diusion proles used in previous models. For practical use, we
suggest a fourth-order polynomial expression for the eddy diusion prole:
Ke (z) =
where
Ke



5 × 103
z<1


e−5.22+15.7z−3.97z2 +0.401z3 −0.0137z4
1 6 z < 10
(3)
is the eddy diusivity in units of
cm2 s−1 ; z
is the altitude in units of 100 km.
Chemical kinetics are examined and modied so as to match the observations of C2 H4 in
the stratosphere. The new expression agrees with the laboratory measurements at all high
temperatures and pressures. This might help to solve a similar problem for the photochemical modeling of C2 H4 on giant planets.
We suggest new laboratory measurements at low
temperature and low pressure to settle this issue.
Acknowledgements
This research was supported in part by the Cassini UVIS program via NASA grant
JPL.1459109 to the California Institute of Technology.
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YLY and XZ were supported in
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part by NASA NNX09AB72G grant to the California Institute of Technology.
We thank
Michael Line for the discussion on the retrieval algorithm and the bootstrap method for
error analysis.
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Table 1: Summary of reaction rate coecients. M represents any third body. Two-body rate constants and
high-pressure limiting rate constants for three-body reactions (k∞ ) are in units of cm3 s−1 . Low-pressure
limiting rate constants for three-body reactions (k0 ) are in units of cm6 s−1 .
Table 2: Photochemical reaction list. See Moses et al. (2005) for reaction rate coecients and references
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Figure 1: Summation of the square of fractional changes over 82 levels and 297 reactions for 18 species when
the rate coecient for each reaction is doubled. For clarity, radicals are not shown in the gure because their
abundances are always not aected by transport due to small chemical destruction timescales. The total
sensitivity is divided into three parts: red part is the contribution from 50 km to 500 km; magenta, from 500
km to 1000 km; yellow, above 1000 km.
Figure 2: Retrieved eddy diusion prole (blue line) plotted with the eect of aerosol heating. Red dashed
line is the temperature prole without aerosol heating; red solid line is the temperature prole with the
aerosol heating by a detached hazed layer (Lavvas et al., 2009).
Figure 3: The eddy diusion prole retrieved from the abundances of C2 H2 and 40Ar (blue sold line) along
with the molecular diusion coecients for CH4 (red line) and the initial guess of the eddy diusion prole
(blue dashed line). Solid black line is the median value of the synthetic set of eddy diusion proles. Shaded
region shows the 1-σ condence level around the median.
Figure 4: Model calculated vertical prole of the species ( 40Ar, C2 H2 ) used for the retrieval of the eddy
diusion prole along with the CH4 prole. Magenta points with error bars are the INMS measurements
(Yelle et al., 2008; Magee et al., 2009). Black dots are the measurements from FUV stellar occultation.
Dashed lines are the modeled proles for C2 H2 from Lavvas et al., 2008b (blue line) and Krasnopolsky, 2009
(basic model: green line, appendix model: cyan line). Red line is our model using the retrieved eddy diusion
prole.
Figure 5: The mixing ratios of the major hydrocarbons in dierent models. Blue dashed line is the result
using empirical eddy diusion prole (Liang et al., 2007); blue solid line is the result using the retrieved
eddy diusion prole; red line is the result using the retrieved eddy diusion prole and with the revised
rate coecients for R109 and R114. The observations are plotted along with the model results. Dark dots
in the upper atmosphere are from FUV stellar occultation (Kammer et al., 2011). Dark dots in the lower
atmosphere are from CIRS limb view (Vinatier et al., 2010). Magenta points at about 1000 km are from
INMS (Magee et al., 2009).
Figure 6: Production and loss rates of C2 H4 plotted in percentage of the total rate. Only the three most
important reactions are plotted for clarity. Dierent reactions are identied by dierent colors according
to the labels in the gure. At each level, the area of a color patch equals to its contribution to the total
production or loss rate.
Figure 7: Main chemical scheme for C2 -hydrocarbons. Yellow arrow indicates photodissociation is involved.
Red arrow indicates three-body reaction is involved. Other reactions are shown in blue.
Figure 8: Color shows the fractional change of C2 H4 when each reaction rate coecient is doubled. The
abscissa is the reaction number in the model. Any reaction with a number larger than 120 is not important
and is omitted for clarity. The 120 reactions are listed in Table ??
Figure 9: Rate coecients for the reaction H + C2 H2 + M −−→ C2 H3 + M. Colored dots with error bars are the
laboratory measurements from Payne and Stief (1976). Temperatures indicated by the color are listed in the
gure. Dashed lines are the reaction rate coecients used in Moses et al. (2005); solid lines are the reaction
rate coecients used in Moses et al. (2000). No laboratory measurements are available for temperatures on
Titan (∼170 K) and in the pressure range of the stratosphere (∼0.1 hpa).
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Figure 10: Rate coecients for the reaction H + C2 H4 + M −−→ C2 H5 + M. Colored dots with error bars
are the laboratory measurements from Lightfoot and Pilling (1987). Temperatures indicated by the color are
listed in the gure. Dashed lines are the reaction rate coecients used in Moses et al. (2005); solid lines are
the new reaction rate coecients (see Table 1). No laboratory measurements are available for temperatures
on Titan (∼170 K) and in the pressure range of the stratosphere (∼0.1 hpa).
Figure 11: Comparison of dierent eddy diusion proles used in previous photochemical modelings: red
(Yelle et al., 2008); green (Lavvas et al., 2008b); blue (Lara et al., 1996); cyan (this work); Dashed line
(molecular diusivity for CH4 ).
22
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table1.PDF
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Reaction
H + C2 H2
+ M −−→ C2 H4 + M
H + C2 H4
+ M −−→ C2 H5 + M
Moses et al. (2000)
k0 = 8.2 × 10−31 e−352/T
k∞ = 1.4 × 10−11 e−1300/T
k0 = 1.3 × 10−29 e−380/T
k∞ = 6.6 × 10−15 T 1.3 e−650/T
Moses et al. (2005)
k0 = 3.3 × 10−30 e−740/T
k∞ = 1.4 × 10−11 e−1300/T
k0 = 1.68 × 10−38 T 2.87 e−923/T
k∞ = 6.6 × 10−15 T 1.3 e−650/T
This work
Moses et al. (2000)
k0 = 5.4 × 10−25 T −1.46 e−1300/T
k∞ = 1.8 × 10−13 T 0.7 e−600/T
1
table2.PDF
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Table 2: Photochemical reaction list.
See Moses et al. (2005) for reaction rate coecients and references.
No.
Reaction No.
Reaction
R1
H2 −→ 2H R61
1, 3 − C4 H6 −→ C4 H5 + H
3
1, 3 − C4 H6 −→ C4 H4 + H2
R2
CH2 −→ CH + H R62
R3
CH3 −→ CH + H2 R63
1, 3 − C4 H6 −→ C3 H3 + CH3
1, 3 − C4 H6 −→ C2 H4 + C2 H2
R4
CH3 −→1 CH2 + H R64
R5
CH4 −→ CH3 + H R65
1, 3 − C4 H6 −→ 2C2 H3
C4 H8 −→ 1, 3 − C4 H6 + 2H
R6
CH4 −→1 CH2 + H2 R66
R7
CH4 −→1 CH2 + 2H R67
C4 H8 −→ C3 H5 + CH3
C4 H8 −→ CH3 C2 H + CH4
R8
CH4 −→3 CH2 + 2H R68
R9
CH4 −→ CH + H + H2 R69
C4 H8 −→ CH2 CCH2 + CH4
R10
C2 H2 −→ C2 H + H R70
C4 H8 −→ C2 H5 + C2 H3
C4 H8 −→ 2C2 H4
R11
C2 H2 −→ C2 + H2 R71
R12
C2 H3 −→ C2 H2 + H R72
C4 H8 −→ C2 H2 + 2CH3
C4 H10 −→ C4 H8 + H2
R13
C2 H4 −→ C2 H2 + H2 R73
R14
C2 H4 −→ C2 H2 + 2H R74
C4 H10 −→ C3 H8 +1 CH2
C4 H10 −→ C3 H6 + CH4
R15
C2 H4 −→ C2 H3 + H R75
R16
C2 H5 −→ CH3 +1 CH2 R76
C4 H10 −→ C3 H6 + CH3 + H
R17
C2 H6 −→ C2 H4 + H2 R77
C4 H10 −→ C2 H6 + C2 H4
C4 H10 −→ 2C2 H5
R18
C2 H6 −→ C2 H4 + 2H R78
R19
C2 H6 −→ C2 H2 + 2H2 R79
C4 H10 −→ C2 H4 + 2CH3
R20
C2 H6 −→ CH4 +1 CH2 R80
C5 H4 −→ C3 H2 + C2 H2
C6 H2 −→ C6 H + H
R21
C2 H6 −→ 2CH3 R81
R22
C3 H2 −→ C3 + H2 R82
C6 H2 −→ C4 H + C2 H
C6 H4 −→ C6 H3 + H
R23
C3 H3 −→ C3 H2 + H R83
R24
C3 H3 −→ C3 H + H2 R84
C6 H4 −→ C6 H2 + H2
R25
CH3 C2 H −→ C3 H3 + H R85
C6 H4 −→ C4 H2 + C2 H2
C6 H5 −→ C6 H4 + H
R26
CH3 C2 H −→ C3 H2 + H2 R86
R27
CH3 C2 H −→1 CH2 + C2 H2 R87
C6 H5 −→ C4 H3 + C2 H2
C6 H6 −→ C6 H5 + H
R28
CH2 CCH2 −→ C3 H3 + H R88
R29
CH2 CCH2 −→ C3 H2 + H2 R89
C6 H6 −→ 3C2 H2
C6 H6 −→ C5 H3 + CH3
R30
C3 H5 −→ CH3 C2 H + H R90
R31
C3 H5 −→ CH2 CCH2 + H R91
C6 H6 −→ C4 H2 + C2 H4
R32
C3 H5 −→ C2 H2 + CH3 R92
C6 H6 −→ 2C3 H3
R33
C3 H6 −→ C3 H5 + H R93
LC6 H6 −→ C6 H5 + H
R34
C3 H6 −→ CH3 C2 H + H2 R94
LC6 H6 −→ C6 H4 + H2
LC6 H6 −→ C5 H3 + CH3
R35
C3 H6 −→ CH2 CCH2 + H2 R95
LC6 H6 −→ C4 H4 + C2 H2
R36
C3 H6 −→ C2 H4 +1 CH2 R96
R37
C3 H6 −→ C2 H3 + CH3 R97
LC6 H6 −→ 2C3 H3
R38
C3 H6 −→ C2 H2 + CH4 R98
C8 H2 −→ C6 H + C2 H
R39
C3 H8 −→ C3 H6 + H2 R99
C8 H2 −→ 2C4 H
R40
C3 H8 −→ C2 H6 +1 CH2 R100
2H + M −→ H2 + M
R41
C3 H8 −→ C2 H5 + CH3 R101
H + CH −→ C + H2
R42
C3 H8 −→ C2 H4 + CH4 R102
H +1 CH2 −→ CH + H2
continued on next page
1
Table 2continued from previous page
No.
Reaction
R43
C4 H2 −→ C4 H + H
R44
C4 H2 −→ C2 H2 + C2
R45
C4 H2 −→ 2C2 H
R46
C4 H4 −→ C4 H2 + H2
R47
C4 H4 −→ 2C2 H2
R48
1 − C4 H6 −→ C4 H4 + 2H
R49
1 − C4 H6 −→ C3 H3 + CH3
R50
1 − C4 H6 −→ C2 H5 + C2 H
R51
1 − C4 H6 −→ C2 H4 + C2 H + H
R52
1 − C4 H6 −→ C2 H3 + C2 H + H2
R53
1 − C4 H6 −→ 2C2 H2 + H2
R54
1, 2 − C4 H6 −→ C4 H5 + H
R55
1, 2 − C4 H6 −→ C4 H4 + 2H
R56
1, 2 − C4 H6 −→ C3 H3 + CH3
R57
1, 2 − C4 H6 −→ C2 H4 + C2 H2
R58 1, 2 − C4 H6 −→ C2 H3 + C2 H2 + H
R59 1, 2 − C4 H6 −→ C2 H3 + C2 H + H2
R60
1, 2 − C4 H6 −→ 2C2 H2 + H2
2
No.
R103
R104
R105
R106
R107
R108
R109
R110
R111
R112
R113
R114
R115
R116
R117
R118
R119
R120
Reaction
H +3 CH2 −→ CH + H2
H +3 CH2 + M −→ CH3 + M
H + CH3 + M −→ CH4 + M
H + CH4 −→ CH3 + H2
H + C2 H + M −→ C2 H2 + M
H + C2 H2 −→ C2 H + H2
H + C2 H2 + M −→ C2 H3 + M
H + C2 H3 −→ C2 H2 + H2
H + C2 H3 + M −→ C2 H4 + M
H + C2 H4 −→ C2 H3 + H2
H + C2 H4 + M −→ C2 H5 + M
H + C2 H5 −→ 2CH3
H + C2 H5 −→ C2 H4 + H2
H + C2 H5 + M −→ C2 H6 + M
H + C2 H6 −→ C2 H5 + H2
H + C3 H2 + M −→ C3 H3 + M
H + C3 H3 + M −→ CH3 C2 H + M
H + C3 H3 + M −→ CH2 CCH2 + M