Icarus 216 (2011) 116–119
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Icarus
journal homepage: www.elsevier.com/locate/icarus
Insolation in Titan’s troposphere
Juan M. Lora a,⇑, Paul J. Goodman b, Joellen L. Russell b, Jonathan I. Lunine a,1
a
b
Lunar and Planetary Laboratory, University of Arizona, 1629 E. University Blvd., Tucson, AZ 85721, USA
Department of Geosciences, University of Arizona, 1040 E 4th St., Tucson, AZ 85721, USA
a r t i c l e
i n f o
Article history:
Received 8 November 2010
Revised 16 August 2011
Accepted 18 August 2011
Available online 31 August 2011
Keywords:
Titan
Radiative transfer
a b s t r a c t
Seasonality in Titan’s troposphere is driven by latitudinally varying insolation. We show that the latitudinal distributions of insolation in the troposphere and at the surface, based on Huygens DISR measurements, can be approximated analytically with nonzero extinction optical depths s, and are not equivalent
to that at the top of the atmosphere (s = 0), as has been assumed previously. This has implications for the
temperature distribution and the circulation, which we explore with a simple box model. The surface
temperature maximum and the upwelling arm of thermally-direct meridional circulation reach the midlatitudes, not the poles, during summertime.
Ó 2011 Elsevier Inc. All rights reserved.
1. Introduction
2. Insolation
Circulation models for the troposphere of Titan mainly focus on
meridional motion as the primary driver for the climate, typically
invoking the upwelling branch of a circulation cell as the source of
convective cloud formation. Two sets of models suggest that all
cloud activity is explained with a simple seasonally-wandering Hadley-type cell that creates a seasonal sinusoidal pattern of upwelling
through a Titan year (Mitchell et al., 2006, 2009; Mitchell, 2008; Tokano et al., 1999; Tokano, 2005, 2009); in cases without methane, this
upwelling arm reaches the pole during summer solstice. These models, however, apply an insolation distribution that ignores any latitudinal dependence of attenuation by the atmosphere, allowing the
daily-average insolation maximum at the surface to reach the polar
latitudes during solstice (see Fig. 1). The optical thickness of Titan’s
atmosphere dictates instead that the surface maxima occur closer to
midlatitudes at solstices. This effect prevents any upwelling arm of a
circulation cell from reaching the poles following the surface
heating.
In this paper, we construct a more physically reasonable distribution of insolation in Titan’s troposphere, taking into account the
attenuation effect described above (Section 2). We explore the
implications of this distribution on surface temperatures and the
thermally direct circulation using a very simple box model (Section
3). We show that our model correctly reproduces certain aspects of
Titan’s seasonality as seen in Cassini–Huygens data (Section 4).
Titan’s hazy atmosphere efficiently absorbs and scatters solar
radiation. Due to the nonzero optical depth of the atmosphere,
the magnitude of insolation versus latitude varies at different levels in the atmosphere. At latitude / and optical depth s, the dailyaverage direct insolation is given by
⇑ Corresponding author.
1
E-mail address: jlora@lpl.arizona.edu (J.M. Lora).
Present address: Department of Astronomy, Cornell University.
0019-1035/$ - see front matter Ó 2011 Elsevier Inc. All rights reserved.
doi:10.1016/j.icarus.2011.08.017
Sav e ¼
S0 E2
2p
Z
h0
ðsin / sin d
h0
þ cos / cos d cos hÞes=ðsin / sin dþcos / cos d cos hÞ dh
ð1Þ
where S0 is the solar constant at Titan, h is the hour angle of the Sun
and d is the solar declination. h0 denotes the hour angle of sunrise/
sunset; the Sun does not set around solstice at the summer pole,
thus h0 is simply p. The factor E = 1 + e cos(H -) is due to Saturn’s
orbital eccentricity e, where H is approximately the angle of the orbit (H = 0 at vernal equinox) and - is Saturn’s longitude of perihelion. As described by Aharonson et al. (2009), this effect causes the
summer solstice solar flux at the top of the atmosphere to differ by
about 1.5 W m2 between southern and northern poles.
At the top of the atmosphere, s = 0 and only curvature affects
the latitudinal distribution of insolation, for a given d. In the atmosphere, where s is nonzero, the airmass encountered by the solar
beam also depends on latitude so that attenuation is greater at
the poles than at low latitudes. Thus, the insolation distribution
at the surface is qualitatively different from that at the top of the
atmosphere for any nonzero value of s.
Tomasko et al. (2008) measured the net solar flux as a fraction
of incident solar flux at multiple levels in the atmosphere with the
DISR instrument aboard the Huygens probe. From these measurements and Eq. (1), we can calculate ‘‘effective’’ extinction optical
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J.M. Lora et al. / Icarus 216 (2011) 116–119
2
Daily average insolation (W/m ): τ = 0
latitude (degrees)
60
4
5
2
5
6
80
1
3
4
40
1
2
2
3
4
3
20
4
4
0
5
−20
3
2
1
4
5
3
2
−40
−60
3
6
1
4
90
6
0
2
5
−80
7
180
270
360
L (degrees)
s
Daily average insolation (W/m2): τ = 0.4
80
0.5
1
1.5
latitude (degrees)
60
2
2.5
0.5
1
0.5
2
40
1
1.5
2.5
1.5
20
2.5
2
2.5
0
2.5
2
3
2.5
−20
2
−40
1.5
1
1.5
2.5
2
1.5
1
0.5
2.5 3
2
3
−60
3.5
0.5
1
0.5
−80
0
2
90
180
270
360
L (degrees)
s
Fig. 1. Daily-averaged insolation versus time. Top: The insolation at the top of Titan’s atmosphere, where s = 0. Here, the maximum average insolation occurs at the poles
during solstices. Bottom: The insolation with s = 0.4, which closely matches the heating distribution near the surface.
−4
Solar Energy vs. Latitude (Southern Solstice)
3
x 10
Heating rates from Tomasko et al. (2008)
τ = 0.87
Adjusted τ = 0.4
Adjusted τ = 0.0
2.5
2
2
6
5
4
1.5
3
1
2
0.5
0
1
−80
−60
−40
−20
0
20
40
60
80
Solar Heating Rate (K/24h)
7
Solar Flux (W/m )
depths for shortwave radiation and, using these optical depths
again in Eq. (1), approximate the daily-average solar flux at all latitudes (see Fig. 1). This approach yields an extinction optical depth
s = 0.87 at the surface. A key feature of these results is that the
maximum in surface insolation does not reach the pole during
summer solstice.
Scattering in Titan’s atmosphere causes diffuse radiation to be a
significant component of the solar flux; thus, the above approximation is not entirely valid. However, Tomasko et al. (2008) computed
solar heating rates as a function of altitude for different latitudes at
different seasons, including a scattering model, and also found that
the maximum (below about 50 km) reached only mid-latitudes
around solstice. That is, in the troposphere, the insolation distribution is qualitatively different from that at the top of the atmosphere,
and is better described by Eq. (1) than by a distribution that ignores
the longer path and greater attenuation that sunlight encounters at
higher latitudes. This distribution with latitude can be approximated with an adjusted extinction optical depth in Eq. (1), normalized to match the DISR measurements of solar flux (Fig. 2). The
resulting insolation distribution for the surface, throughout a Titan
year, is shown in the bottom panel of Fig. 1.
0
Latitude
Fig. 2. Calculations of the solar energy at the surface around solstice using Eq. (1).
The s = 0.0, 0.4 curves are adjusted to match the DISR measurements at around 10°S
latitude. Note that the vertical axis for the heating rates from Tomasko et al. (2008)
(solid curve) is on the right. The best match to these heating rates is the adjusted
s = 0.4 curve; the worst match is s = 0.
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J.M. Lora et al. / Icarus 216 (2011) 116–119
3. Implications for surface temperatures and circulation
We can examine the basic response of the surface and troposphere to the insolation distributions discussed in Section 2. In particular, we consider the differences between the top-ofatmosphere (zero-opacity) distribution, with flux values scaled
for the lower atmosphere and surface, and the nonzero-opacity
distribution (Fig. 1, bottom) with s = 0.40 at the surface.
3.1. Methods
We implement a simple grey atmosphere box model of Titan’s
troposphere as an order-of-magnitude study of the implications
of these insolation distributions. The model consists of twelve
‘‘boxes,’’ six upper and six lower, which represent Titan’s troposphere (with an additional six boxes for the surface), in which temperature, T, is solved via the thermodynamic equation:
Qcp T Dh
Vcp qT @h
¼ AðS F outIR þ F inIR Þ þ
þ SH
h @t
h
ð2Þ
where
h¼T
j
P0
P
Q ¼ kqDhbottom
SH ¼ cp qus f ðT air T surf Þ
ð3Þ
SH is the sensible heat term (heat exchange between surface and
bottom atmospheric boxes), and Q parameterizes advection. All
other model parameters are listed in Table 1. The temperature evolution of the surface follows a similar function to Eq. (2), but without the advection term.
To compute solar heating rates for each box, the insolation distributions are averaged over the appropriate latitudes (i.e. 90°S to
60°S for the southern polar box). In the case of the nonzero distribution, s = 0.32, 0.35, and 0.40 for the top, bottom, and surface
boxes, respectively. Each box also emits thermally according to
FIR = rT4, with constant emissivities.
Advection is parameterized assuming only a thermally direct
circulation. In each group of four adjacent atmospheric boxes, the
direction of flow is governed by the direction in which the warmest
air ascends; the magnitude is determined by the temperature
difference Dhbottom between the two bottom boxes, as a proxy for
buoyancy. The total advection in each box is the sum of the
Table 1
List of parameters and constants.
Parameter
Description
Units
V
T
h
Volume of box
Temperature
Potential temperature
Air density
Area of box side
Absorbed shortwave radiation
Infrared radiation flux
Pressure
Advection parameter
Surface wind term
Emissivity
m3
K
K
kg m3
m2
W m2
W m2
Pa
m3 s1 K1
m s1
Dimensionless
q
A
S
FIR
P
k
us
Constant
Description
Value
cp
Heat capacity
r
Stefan–Boltzmann constant
Reference pressure
Poisson constant
Drag coefficient
103 J kg1 K1 (air)
107 J m2 K1 (surfacea)
5.67 108 W m2 K4
1450 hPa
0.3
0.002
P0
j
f
a
From Flasar (1998).
advection in both adjacent cells (except the polar boxes, which
only take part in one cell).
It should be noted that the box model only considers thermally
driven advection, which assumes a significant seasonal response to
heating. Because of Titan’s slow rotation, the mean meridional circulation is expected to extend to the poles (Mitchell et al., 2006);
thus, we ignore the effects of dynamics on the latitudinal extent
of upwelling and downwelling. Mitchell et al. (2006) also showed
that latent heat release could limit the poleward extent of upwelling; here, we do not consider methane thermodynamics and concentrate exclusively on the response due to solar heating.
3.2. Results
Under the zero-opacity insolation distribution, the calculated
surface temperature maximum reaches the polar region during
solstice, making the summer pole warmer than the equatorial surface by nearly 3 K, and creating a pole-to-pole temperature gradient. The resulting circulation has upwelling and downwelling in
the summer and winter hemispheres, respectively (Fig. 3a). Around
equinox, the temperature maximum occurs at the equator (Fig. 4),
and the circulation’s upwelling branch shifts hemispheres, following the pole-to-pole sinusoidal pattern of the insolation.
When forced with the nonzero-opacity insolation distribution,
the surface temperature maximum oscillates between northern
and southern mid-latitude boxes. The surface temperature difference between equator and pole remains positive year-round
(Fig. 4), and is approximately 1 K and up to 5 K in the summer
and winter hemispheres, respectively, in agreement with observations (Jennings et al., 2009). Additionally, the surface temperature
at the pole is about 1 K warmer in southern summer than northern
summer. Upwelling in the model remains at low-and mid-latitudes
throughout the year, reaching the mid-latitude boxes during summer, with little movement in the polar box (Fig. 3b). The strongest
upwelling remains at equatorial latitudes year-round. Summer
upwelling at southern equatorial and mid-latitude boxes is stronger than that in the north, due to the effect of the orbital eccentricity on the insolation.
4. Discussion and conclusions
Observations of clouds on Titan suggest an active and important
tropospheric circulation, with significant activity at the summer
pole (Brown et al., 2002, 2010; Porco et al., 2005; Rodriguez
et al., 2009; Turtle et al., 2009, 2011a). A realistic treatment of
the attenuation of sunlight in the lower atmosphere, however,
indicates that the maximum insolation does not occur over the
pole during solstice, posing an obstacle to previous explanations
of polar clouds.
Our simple model results highlight the implications of this insolation distribution: Assuming a completely homogeneous surface,
the highest surface temperatures would oscillate between mid-latitudes, not the poles, with season, following the maximum insolation. This is in agreement with CIRS measurements of temperature
during mid to late southern summer, where the equator was warmer than the summer pole by about 2 K (Jennings et al., 2009). The
upwelling arm of a circulation cell (assumed here to be a proxy for
possible cloud formation) following this pattern of heating would
thus be confined to the mid- and low-latitudes instead of wandering between poles. Still substantial heating of the summer pole,
which might be relatively humid due to surface liquids, could still
produce convective cloud formation, but in a region of otherwise
little vertical motion. Considerable upwelling around the equator
during equinox would be consistent with the most recent cloud
observations (Turtle et al., 2011a,b), as well as previous models
(Mitchell et al., 2006, 2009; Tokano, 2009).
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J.M. Lora et al. / Icarus 216 (2011) 116–119
100
b
Pressure (hPa)
a
Pressure (hPa)
100
1000
1000
1450
1450
South
Equator
North
South
Equator
North
Temperature (K)
Fig. 3. Box model results for southern summer: (a) zero-opacity insolation distribution; (b) nonzero-opacity insolation distribution. Temperatures (K) are shown in color;
arrows show the circulation. In the first case, atmospheric and surface emissivities were about 0.6 and 0.7, with k 1016 m3 s1 K1; in the second, emissivities were 0.85 and
0.95, with k about half the previous value.
98
potentially constraining the poleward extent of tropospheric
upwelling.
96
Acknowledgments
The authors thank M. Tomasko and his coauthors for their calculations of atmospheric heating rates, as well as J. Mitchell and an
anonymous reviewer for important and helpful comments. This
work was funded by the Cassini Project.
94
92
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88
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0
90
180
270
360
Ls (degrees)
Fig. 4. The evolution of surface temperature for the zero-opacity (thin lines) and
nonzero-opacity (thick lines) insolation distributions. South polar (solid) and
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