1
JOVIAN STRATOSPHERE AS A CHEMICAL TRANSPORT
2
SYSTEM: BENCHMARK ANALYTICAL SOLUTIONS
3
4
XI ZHANG1, RUN-LIE SHIA1, AND YUK L. YUNG1
5
6
7
1
8
Pasadena, CA, 91125, USA; xiz@gps.caltech.edu
Division of Geological and Planetary Sciences, California Institute of Technology,
9
10
11
12
13
14
15
16
17
To be submitted to
18
The Astrophysical Journal
19
20
21
22
23
24
25
26
27
28
29
30
31
1
32
ABSTRACT
33
34
We systematically investigated the solvable analytical benchmark cases in both one- and
35
two-dimensional (1-D and 2-D) chemical-advective-diffusive systems. We use the
36
stratosphere of Jupiter as an example but the results can be applied to other planetary
37
atmospheres and exoplanetary atmospheres. In the 1-D system, we show that CH4 and
38
C2H6 are mainly in diffusive equilibrium, and the C2H2 profile can be approximated by
39
the modified Bessel functions. In the 2-D system in the meridional plane, analytical
40
solutions for two typical circulation patterns are derived. Simple tracer transport
41
modeling demonstrates that the distribution of a short-lived species (such as C2H2) is
42
dominated by the local chemical sources and sinks, while that of a long-lived species
43
(such as C2H6) is significantly influenced by the circulation pattern. We find an equator-
44
to-pole circulation could qualitatively explain the Cassini observations, but a pure
45
diffusive transport process could not. For slowly rotating planets like the close-in
46
extrasolar planets, the interaction between the advection by the zonal wind and chemistry
47
might cause a phase lag between the final tracer distribution compared and the original
48
source distribution. The numerical simulation results from the 2-D Caltech/JPL
49
chemistry-transport model agree well with the analytical solutions for various cases.
50
51
52
53
54
55
56
57
58
59
60
61
62
2
63
64
1. INTRODUCTION
65
66
The Jovian stratosphere is an ideal laboratory to study atmospheric tracer transport. The
67
stratosphere is dominated by the hydrocarbon photochemistry, driven by the photolysis of
68
the parent species, methane (CH4), which is transported from the deep atmosphere. Two
69
most abundant photochemical products, acetylene (C2H2) and ethane (C2H6), have
70
properties to make them ideal tracers. First, besides CH4, they show the most prominent
71
features in the middle infrared emission spectra of Jupiter. Therefore their latitudinal and
72
vertical distributions can be accurately determined. Second, their chemical lifetimes are
73
different by about two orders of magnitude, ranging from several Earth years (C2H2) to
74
several hundreds of Earth years (C2H6). That means they have different sensitivity to the
75
transport. In fact, their latitudinal profiles (Nixon et al. 2007) show opposite trends,
76
implying that the transport timescale is probably located within the two lifetimes. Third,
77
their chemistry is relatively simple and most of the chemical reaction coefficients have
78
been measured in the laboratory with small uncertainties. Unlike the other possible
79
tracers, such as hydrogen cyanide (HCN) and carbon dioxide (CO2), whose vertical
80
distribution is not known (Lellouch et al. 2006), or aerosol, which might be affected by
81
complicated microphysics, the simple combination of C2H2 and C2H6 contains a wealth
82
of information of the stratospheric circulation on Jupiter.
83
84
Most of the previous studies focused on 1-D chemistry-diffusion models (e.g., Strobel
85
1974; Gladstone et al. 1996; Moses et al. 2005), which essentially ignore the latitudinal
86
transport. The advantages of 1-D model are: (1) it is numerically stable due to the nature
87
of diffusive processes; (2) the computation is usually fast, and therefore it could include a
88
very complicated network of chemical reactions (Moses et al. 2005). Once the horizontal
89
and vertical advection terms are added, the model is subject to the numerical instability
90
and limited by the Courant-Friedrichs-Lewy (CFL) criterion, although the 2-D
91
calculation is more realistic.
92
3
93
There is no definitive 2-D CTM for the stratosphere of Jupiter, taking into account the
94
photochemistry, eddy and molecular diffusion, as well as the vertical and horizontal
95
advection, although the existence of large-scale stratospheric circulation has been
96
hypothesized since the 1990s (e.g., Conrath et al. 1990; West et al. 1992). Friedson et al.
97
(1999) proposed that the horizontal eddy mixing processes dominate the transport of the
98
SL9 debris in the stratosphere of Jupiter. Liang et al. (2005) used a 2-D chemistry-
99
diffusion model and found that the horizontal mixing might be enough to explain the
100
latitudinal profiles of C2H2 and C2H6. A simple 1-D model in the latitudinal coordinate by
101
Lellouch et al. (2006) shows that the dynamical pictures derived from HCN and CO2 are
102
not consistent with each other, and also not with the C2H2 and C2H6 profiles. Both Liang
103
et al. (2005) and Lellouch et al. (2006) suggested that the horizontal eddy diffusivity is
104
required to vary with latitude and altitude, leading to a more complicated picture. Note
105
that the C2H6 distribution cited in their studies is decreasing from low latitudes to high
106
latitudes. The recent analysis of Cassini and Voyager spectra has revealed more accurate
107
latitudinal profiles of C2H6 (Nixon et al. 2010; Zhang et al. 2012a), which are clearly
108
enhanced in the high latitudes, especially in the Voyager era. One might also use a
109
latitudinally varying vertical eddy diffusivity profile to explain the C2H6 horizontal
110
distribution via changing its vertical slope with latitude (Lellouch et al. 2006). However,
111
this approach might be no different to a parameterization of a realistic horizontal and
112
vertical advection process. Instead, a full CTM is needed to understand the tracer
113
transport in the stratosphere of Jupiter.
114
115
As mentioned above, a very careful treatment in the numerical scheme is necessary in the
116
CTM since the advection terms might lead to less accurate results. Shia et al. (1990)
117
compared different numerical schemes and adopted the modified Prather scheme (Prather
118
1986) in the Caltech/JPL Kinetics CTM. In that paper, the authors derived several
119
analytical solutions to validate the numerical results, in both 1-D and 2-D. But the authors
120
only used the analytic solutions to test the numerical scheme and did not discuss the
121
underlying physical implications of those analytical results. Therefore, some of their
122
analytical results were only mathematically correct but physically counterintuitive (such
123
as the negative chemical production rate, etc.).
4
124
125
On the other hand, the nonlinear feedbacks in the complicated chemical-advective-
126
diffusive system may blur the physical insights. A simple but realistic analytical solution
127
can be considered as a benchmark case for understanding the basic behavior of the
128
system, under idealized assumptions. Previous studies did not focus on the analytical
129
benchmark cases in the atmospheric tracer transport. In civil engineering, the regional
130
Gaussian-plume dispersion models have been studies for many years, and the analytical
131
solutions for the three-dimensional (3-D) diffusion equation could be obtained, although
132
they may not be in the explicit forms (e.g., Lin and Hildemann, 1997). But those
133
solutions are not useful for this study because (1) they are too complicated to show any
134
physical insight; (2) they are restricted to a nonreactive contaminant; and (3) they are not
135
in the planetary scale in which the sphericity of the planet should be taken into account.
136
For the simple planetary-scale analytical solutions, besides Shia et al. (1990), previous
137
attempts mainly focused on the 1-D solutions. Neglecting the chemistry, Chamberlain
138
and Hunten (1987) derived the 1-D analytical solution with an exponential form of eddy
139
and molecular diffusivities. Yelle et al. (2001) reported a 1-D diffusive equilibrium CH4
140
profile, which is essentially a special case of that by Chamberlain and Hunten (1987). A
141
systematic study of the available analytical cases in the planetary chemical-transport
142
system has been lacking.
143
144
In this study we systematically investigate the behavior of the chemical-advective-
145
diffusive system through various representative analytical benchmark cases, such as for
146
the long-lived species versus the short-lived species. Those analytical formulas will be
147
used to validate the numerical simulations in which the numerical schemes are usually
148
not trivial. We will focus on the hydrocarbons in the stratosphere of Jupiter because the
149
observations of C2H2 and C2H6 show a beautiful example of the tracer transport systems.
150
In order to derive the analytical formulas, we need to make some simplifying
151
assumptions; therefore, we will leave the detailed numerical modeling with realistic
152
hydrocarbon chemistry and circulation pattern inferred from the radiative modeling
153
(Zhang et al. 2012a, 2012b) to a future study. Finally, our results could be applied to
154
other planetary and exoplanetary atmospheres.
5
155
156
This paper is structured as follows. In section 2, we will introduce the chemical-
157
advective-diffusive equation. In section 3, we will solve the equation in the 1-D system.
158
In sections 4 and 5 we will focus on the 2-D systems in the meridional plane and zonal
159
plane, respectively, followed by a summary in section 6.
160
161
2. THE NATURE OF THE PROBLEM
162
163
Let us first consider a chemical system in a fast rotating atmosphere. Every quantity can
164
be zonally averaged. We adopt the Transformed Eulerian Mean (TEM) formulation
165
(Andrews et al. 1987, hereafter AHL1987) here. Chemical species are transported
166
vertically and meridionally by the residue mean circulation driven by the diabatic
167
circulation, with a vertical effective transport velocity 𝑤 and a meridional effective
168
transport velocity 𝑣. We also parameterize the eddy transport in a “diffusion” tensor that
169
governs the tracer mixing processes both vertically and meridionally (See AHL1987,
170
p.354). In the region above the homopause, species with different mass would separate
171
from each other by molecular diffusion.
172
173
We adopt a vertical coordinate 𝑧 = 𝐻 𝑙𝑛(𝑝𝑠 /𝑝) , where 𝑝 is pressure and 𝑝𝑠 is the
174
reference pressure, which is usually taken to be 1 bar for giant planets. 𝐻 is the pressure
175
scale height of the background atmosphere. The meridonal coordinate is 𝑦 = 𝑎𝜃, where 𝑎
176
is planetary radius, and 𝜃 is the latitude. We further define a dimensionless coordinate
177
𝜉 = 𝑧/𝐻. The volume mixing ratio of gas (or tracer) 𝑖 is 𝜒 = 𝑁𝑖 /𝑁, where 𝑁𝑖 and 𝑁 are
178
the concentrations of gas and background atmosphere, respectively. Below the
179
homopause, the full form of zonal-averaged Eulerian mean transport equation for 2-D
180
chemical system is (Shia et al. 1990)
181
𝜕𝜒
𝜕𝜒
𝜕𝜒
1 𝜕
𝜕𝜒
𝜕
𝜕𝜒
𝑃−𝐿
+𝑣
+𝑤
−
(𝑐𝑜𝑠𝜃 𝐾𝑦𝑦 ) − 𝑒 𝜉 (𝑒 −𝜉 𝐾𝑧𝑧 ) =
.
𝜕𝑡
𝜕𝑦
𝜕𝑧 𝑐𝑜𝑠𝜃 𝜕𝑦
𝜕𝑦
𝜕𝑧
𝜕𝑧
𝑁
182
where 𝑃 and 𝐿 are the chemical source and loss terms, respectively. Here we use only the
183
diagonal term of the diffusion tensor 𝐾. This is particularly an advantage of the TEM
184
formulism since the diabatic circulation has already taken into account the y-z direction
6
(1)
185
transport so that we can neglect the 𝐾𝑦𝑧 and 𝐾𝑧𝑦 terms (AHL 1987, p.380). Above the
186
homopause, strictly speaking, we should also consider the molecular diffusion. However,
187
the transport by the residue circulation is usually more effective in the region where eddy
188
mixing dominates, so we just simply neglect it above the homopause in the 2-D systems.
189
We will consider molecular diffusion in the 1-D system (section 3).
190
191
For the numerical simulation, we use the Caltech/JPL kinetics model. The 1-D model is
192
taken from the state-of-the-art chemical schemes for Jovian stratosphere from Moses et
193
al. (2005). The model integrates the continuity equation including chemistry and vertical
194
diffusion using a matrix inversion method (Allen et al. 1981). We simplify the Moses et
195
al. (2005) model by assuming an isothermal atmosphere and using a simplified molecular
196
and eddy diffusivity profiles, with the chemistry only including the C2 hydrocarbons.
197
This idealized model is indeed very close to the full chemistry model. Fig. 1 shows the
198
numerical results compare with the full chemistry model from Moses (2005), a reduced
199
C2 chemistry model with realistic chemistry and diffusivity, and our idealized model. The
200
idealized model we introduced above agrees well with the full chemistry model. The
201
simplified eddy diffusivity and molecular diffusivity are shown in Fig. 2. For the 2-D
202
simulations in the meridional plane, we adopt the numerical model from Shia et al.
203
(1990) for a single tracer. The details will be referred to section 4.
204
205
3. 1-D SYSTEM
206
207
Consider a 1-D chemical-transport system in the vertical coordinate in the global-average
208
sense. Above the homopause, the vertical diffusive flux 𝜙𝑧 = 𝑁𝐾𝑧𝑧 𝜕𝜒/𝜕𝑧 needs to be
209
modified to include moleculardiffusion
210
𝜙𝑧 = 𝑁𝐾𝑧𝑧
𝜕(𝑁𝑖 /𝑁𝑒𝑞,𝑖 )
𝜕𝜒
+ 𝑁𝐷𝑖
,
𝜕𝑧
𝜕𝑧
(2)
211
where 𝐷𝑖 is the molecular diffusivity for gas component 𝑖 in the background atmosphere,
212
𝑁𝑒𝑞,𝑖 is the equilibrium density profile with the scale height of species 𝑖. After some
213
manipulation, the continuity equation becomes
214
𝜕𝜒
𝜕𝜒
𝜕
𝜕𝜒 𝑓𝐷𝑖
𝑃−𝐿
+𝑤
− 𝑒 𝜉 {𝑒 −𝜉 [(𝐾𝑧𝑧 + 𝐷𝑖 )
+
𝜒]} =
.
𝜕𝑡
𝜕𝑧
𝜕𝑧
𝜕𝑧
𝐻
𝑁
7
(3)
215
where 𝑓 = 𝑚𝑖 /𝑚 − 1 , 𝑚𝑖 and 𝑚 are the molecular mass of the species 𝑖 and the
216
background atmosphere, respectively. In order to derive analytical solutions, we assume
217
some forms for eddy diffusivity and molecular diffusivity. Lindzen (1981) proposed a
218
wave-breaking turbulent mixing diffusivity, which satisfies 𝐾𝑧𝑧 ∝ 𝑁 −1/2 . The binary
219
molecular diffusion theory implies 𝐷𝑖 ∝ 𝑁 −1 (Chamberlain and Hunten, 1987).
220
Therefore, in this study we assume 𝐾𝑧𝑧 = 𝐾0 𝑒 𝛾𝜉 and 𝐷𝑖 = 𝐷0 𝑒 𝜉 .
221
222
For an isothermal atmosphere which approximates Jovian stratosphere, we have 𝑁 =
223
𝑁0 𝑒 −𝜉 . For the chemical production and loss terms, we assume 𝑃 = 𝑃0 𝑁0 𝑒 𝛼𝜉 , and 𝐿 =
224
𝐿0 𝑁𝑒𝛽𝜉 𝜒=𝐿0 𝑁0 𝑒 (𝛽−1)𝜉 𝜒. For a nondivergent flow we take 𝑤 = 𝑤0 𝑒 𝜉 ∝ 𝑁 −1 . For steady
225
state with 𝜕𝜒/𝜕𝑡 = 0 in the vertical coordinate 𝜉, equation (3) becomes
226
[𝐷0 +𝐾0 𝑒 (𝛾−1)𝜉 ]
227
+ 𝑃0 𝐻 2 𝑒 𝛼𝜉 = 0.
228
Equation (4) is the governing equation for the 1-D chemical-advective-diffusive system.
229
There is no general solution for this equation except under some specific conditions. If
230
𝛾 = 𝛽 = 1, we could obtain an analytical solution by following the derivation of Shia et
231
al. (1990). If 𝛽 = 1 and 𝑃0 = 0 , there could be a solution expressed by the
232
hypergeometric functions. Alternatively, in our idealized model, we consider the cases
233
with 𝛾~0.5, which is based on Lindzen’s hypothesis and also approximates the situation
234
in Jovian stratosphere (Moses et al. 2005). For Jupiter, we take 𝑇0 = 150𝐾 , 𝐻 =
235
24.1 𝑘𝑚 , 𝐾0 ~280 𝑐𝑚2 𝑠 −1 , and 𝐷0 ~0.04 𝑐𝑚2 𝑠 −1 for CH4 and 0.03 𝑐𝑚2 𝑠 −1 for C2H2
236
and C2H6 (scaled by the square root of molecular mass). Again, the results from this
237
idealized model are very close to those from the state-of-art Jupiter model (Moses et al.
238
2005), as shown in Fig. 1.
𝑑2𝜒
𝑑𝜒
+ [𝑓𝐷0 − 𝑤0 𝐻+𝐾0 (𝛾 − 1)𝑒 (𝛾−1)𝜉 ]
− 𝐿0 𝐻 2 𝑒 (𝛽−1)𝜉 𝜒
2
𝑑𝜉
𝑑𝜉
(4)
239
240
In principle, it is not proper to include vertical wind in the 1-D model because it will go
241
to infinity when the atmospheric density drops to zero at the top boundary. However, if
242
we artificially add wind in the 1-D case, it is also useful to roughly estimate the effect of
243
vertical transport in the 2-D case. Therefore, we derived the solutions for both wind-free
244
( 𝑤0 = 0) and wind cases ( 𝑤0 ≠ 0 ). The solutions of the 1-D chemical system are
8
245
summarized in Table 1. The detailed derivations can be found in Zhang (2012, Chapter
246
V, PhD thesis).
247
248
3.1 Cases without Wind
249
250
If we set 𝑤0 = 0, the stratosphere of Jupiter can be approximated in a global-average
251
sense. Three typical cases are used to explain the distributions of CH4, C2H6 and C2H2 in
252
the Jovian stratosphere.
253
254
3.1.1 CH4
255
256
CH4 is transported upward from the interior. If we neglect photolysis, CH4 will be
257
governed by the diffusion equilibrium, corresponding to the case I in Table 1. This is
258
generally true because the strong self-shielding effect will limit its photolysis efficiency
259
below some pressure level. The upward flux is on the order of 109 𝑐𝑚2 𝑠 −1 , so 𝐹𝐻/
260
𝑓𝑁0 𝐷0 is ~10−5. Compared with the CH4 mixing ratio in the deep interior, determined by
261
the thermochemistry (𝜒0 ~1.8 × 10−3 ), the flux term can be ignored. Rewrite the solution
262
as
𝑓
𝐷0 𝑒 (1−𝛾)𝜉 +𝐾0 𝛾−1
𝜒(𝜉) = 𝜒0 (
) .
𝐷0 +𝐾0
263
(5)
264
Fig. 3 shows the profile for CH4 (𝑓~6). We can see that the analytical solution matches
265
the numerical model very well; in the lower atmosphere, where 𝐷0 ≪ 𝐾0, it behaves as a
266
constant mixing ratio profile; and in the upper atmosphere, where 𝐷0 ≫ 𝐾0 and the
267
pressure 𝑝 ∝ 𝑒 −𝜉 , it behaves as 𝜒 ∝ 𝑝 𝑓 .
268
269
3.1.2 C2H6
270
271
On the other hand, Jupiter’s C2H6 is formed around the homopause region and
272
transported downward. Therefore the flux term cannot be ignored. Interestingly, the flux
273
is also on the order of 109 𝑐𝑚2 𝑠 −1, so 𝐹𝐻/𝑓𝑁0 𝐷0 is ~10−5 . For C2H6, 𝑓~12. Since the
9
274
source of C2H6 is in the upper atmosphere, we can set the lower boundary condition as
275
𝜒0 = 0, so the solution becomes
𝑓
276
𝐹𝐻
𝐷0 𝑒 (1−𝛾)𝜉 +𝐾0 𝛾−1
𝜒(𝜉) =
[1 − (
) ].
𝑓𝑁0 𝐷0
𝐷0 +𝐾0
277
Fig. 4 shows that the analytical solution matches the model result very well below the
278
homopause. In the lower atmosphere, where 𝐷0 ≪ 𝐾0 , we take the Taylor expansion of
279
the solution and obtain
𝐹𝐻(𝑒 (1−𝛾)𝜉 − 1)
𝜒(𝜉, 𝐷0 ≪ 𝐾0 ) =
,
𝐾0 𝑁0 (1 − 𝛾)
280
(6)
(7)
281
which is consistent with the solution with 𝐷0 = 0 (case II). The solution implies that the
282
C2H6 mixing ratio profile should asymptotically behave as 𝜒 ∝ 𝑝(𝛾−1) (Fig. 4).
283
284
Above the source region the flux changes sign (upward), but since the flux drops fast, we
285
can still ignore it, and the following analytical solution still matches the model result
286
well, similar to the condition of CH4. Let 𝐹 = 0 in the solution of case I and note that at
287
the homopause 𝐷0 𝑒 (1−𝛾)𝜉 = 𝐾0 ,
𝑓
𝐷0 𝑒 (1−𝛾)𝜉 +𝐾0 𝛾−1
𝜒(𝜉) = 𝜒ℎ [
] ,
2𝐾0
288
289
(8)
where 𝜒ℎ is the volume mixing ratio at the homopause.
290
291
Therefore, we conclude that in the Jovian stratosphere CH4 and C2H6 are mostly in
292
diffusive equilibrium, especially in the region where transport is much faster than the
293
chemical processes.
294
295
3.1.3 C2H2
296
297
Above the homopause, the C2H2 profile can be approximated by the diffusive equilibrium
298
(case I, with 𝐹 = 0), as we showed for the C2H6 profile above the homopause. It agrees
299
very well with the numerical simulations (Fig. 5). In the eddy diffusion dominated region,
300
the solution of case III is a good approximation for the vertical profile of C2H2. C2H2 is
10
301
transported downward, with a major chemical loss by combining with a hydrogen atom to
302
form C2H3. This is a three-body reaction so the rate is given by 𝐿 = 𝑘[𝐶2 𝐻2 ][𝐻]𝑁, where
303
[𝐻] is the number density of hydrogen atom. In fact the product 𝑘[𝐻]𝑁 is approximately
304
constant through the lower region (~108 𝑐𝑚−3 𝑠 −1). Therefore 𝛽 is roughly 0 for C2H2.
305
This is not surprising because the major sources of hydrogen atoms are from (1) C2H2
306
photolysis directly; (2) C2H+H2; (3) C2+H2. Note that the latter two reactions are actually
307
driven by C2H2 photolysis as well. So the production rate of hydrogen atom is
308
proportional to the C2H2 abundance. On the other hand, the loss of hydrogen atom is also
309
through combining with C2H2 and C2H3, these reaction rates are also roughly
310
proportional to C2H2 and background atmospheric abundance. Therefore by equating the
311
sources and sinks of the hydrogen atom, the product 𝑘[𝐻]𝑁 can be expressed in terms of
312
several reaction constants and therefore is roughly a constant.
313
314
In the solution of case III, we assume 𝛽~0 and 𝐿0 ~𝑘[𝐻]𝑁~108 𝑐𝑚−3 𝑠 −1, and we obtain
315
2𝐻√𝐿0 /𝐾0 ~30. We have to ignore the 𝐼𝜈 term because the 𝜒(𝜉) is expected to increase
316
with altitude for the source region above. In this case, we have 𝜈 = |𝛽−𝛾| = 1. $$$ The
317
analytical C2H2 profile, as shown in Fig. 5, is
318
𝜒(𝜉) = 𝐶1 𝑒
1−𝛾
(1−𝛾)𝜉
2𝐻√𝐿0 /𝐾0 (𝛽−𝛾)𝜉
2 𝐾𝜈 (
𝑒 2 )
|𝛽 − 𝛾|
𝜉
𝜉
= 𝐶1 𝑒 4 𝐾1 (15𝑒 4 ),
(9)
319
The profile qualitatively agrees with the model result, although not good as the CH4 and
320
C2H6 cases in section 3.1.1 because we simplified the chemistry. But we conclude that
321
the C2H2 profile on Jupiter can be approximated by the modified Bessel function 𝐾𝜈 .
322
323
3.2 Cases with Wind
324
𝑤0 𝐻
325
If we define a new “mass factor” 𝑓 ∗ = 𝑓 −
326
free case that we discussed in Section 3.1. The physical meaning of the correct factor
327
𝑤0 𝐻/𝐷0 is the ratio of molecular diffusive timescale to the vertical advection time scale.
328
Naively we can imagine an upward wind tends to make the gas molecule “lighter,” while
329
a downward wind will make the species “heavier.” This result can be directly applied to
𝐷0
11
, equation (7) will be reduced to the wind-
330
CH4, as shown in Fig. 6, with 𝑤0 = ±5 × 10−8 𝑐𝑚 𝑠 −1 . Since it is not proper to put the
331
wind into numerical simulations, as it will cause numerical problems. We only show the
332
analytical solutions for qualitative illustration. The results show that the upward wind
333
will lift up the homopause and downward wind will push it down. It actually transports
334
the more (less) abundant species from below (above) and thus increases (decreases) the
335
mixing ratio of CH4.
336
337
Since in the 2-D simulation we generally care about is the region below the homopause
338
and also most of the latitudinal observations are located around 1 mbar or below for
339
Jupiter, we present solutions with only eddy diffusivity and wind transport. If we neglect
340
the chemical production and loss terms, the solution (case VII) would have a different
341
slope from the solutions in the wind-free cases. The solution depends on the ratio of
342
diffusion timescale to the advection timescale, 𝑤0 𝐻/𝐾0 . Fig. 7 shows two typical vertical
343
profiles of C2H6, with an upward wind and downward wind (~𝑤0 = ±1 × 10−6 𝑐𝑚 𝑠 −1 )
344
respectively, compared with the wind-free theoretical curve. It shows an upward wind
345
will increase the mixing ratio in the higher altitude by preventing it from being
346
transported downward but a downward wind tends to lower down the mixing ratio of
347
C2H6. Therefore, if we neglect the horizontal transport, the rising air at the equator and
348
sinking air at the poles will result in more C2H6 at equator and less C2H6 at poles, which
349
is opposite to the CIRS measurements (Zhang 2012). $$$ zhang 2012 is Thesis $$$ That
350
suggests the horizontal transport is actually important and a “pseudo-2D” photochemical
351
model is not sufficient to explain the observations. When we add the chemical
352
source/loss, the equation cannot be solved analytically in general, unless we neglect the
353
eddy diffusion term and assume 𝛼 = 𝛽 − 1 and 𝛽 ≠ 0, the solution is shown in case VIII
354
in Table 1.
355
356
4. 2-D SYSTEM IN THE MERIDIONAL PLANE
357
358
Consider a 2-D chemical-advective-diffusive system below the homopause in the
359
latitudinal and vertical coordinate. We still assume the atmosphere to be isothermal and
360
barotropic. From previous experience, we note that the equation with eddy diffusion
12
361
(𝛾~0.5), advection and chemistry all together cannot be solved analytically, even in the
362
1-D case. Therefore we are trying to decouple the processes. We need to introduce the
363
streamfunction 𝜓 so that
364
𝑣=−
365
1
𝜕
𝑒 𝜉 (𝑒 −𝜉 𝜓),
cos 𝜃 𝜕𝑧
1 𝜕𝜓
𝑤=
.
cos 𝜃 𝜕𝑦
(10𝑎)
(10𝑏)
366
367
4.1 Without Chemistry, 𝑷𝟎 = 𝑳𝟎 = 𝟎
368
369
370
If the chemistry can be ignored, the steady state of equation (1) becomes
𝑣
𝜕𝜒
𝜕𝜒
1 𝜕
𝜕𝜒
𝜕
𝜕𝜒
+𝑤
−
(cos 𝜃 𝐾𝑦𝑦 ) − 𝑒 𝜉 (𝑒 −𝜉 𝐾𝑧𝑧 ) = 0.
𝜕𝑦
𝜕𝑧 cos 𝜃 𝜕𝑦
𝜕𝑦
𝜕𝑧
𝜕𝑧
(11)
371
If we also neglect the eddy diffusion term, the solution would be trivial: for any given
372
streamfunction 𝜓, the solution is 𝜒(𝜃, 𝜉) = 𝐺(𝑒 −𝜉 𝜓(𝜃, 𝜉)), where 𝐺 is any functional
373
form determined by the boundary conditions and 𝑒 −𝜉 𝜓 is the mass-weighted
374
streamfunction. On the other hand, by ignoring the advection term, we have a 2-D
375
diffusion equation. But this solution will be trivial too because there is no horizontal
376
diffusion flux without chemical source. It will be reduced to the 1-D case.
377
378
4.2 With Chemistry
379
380
Let us introduce the chemistry. The source and sink terms are parameterized as:
381
𝑃(𝜃, 𝜉) = 𝑃0 𝑁0 𝑒 𝛼𝜉 𝑓(𝜃) , and 𝐿(𝜃, 𝜉) = 𝐿0 𝑁𝑒𝛽𝜉 𝑔(𝜃)𝜒 = 𝐿0 𝑁0 𝑒 (𝛽−1)𝜉 𝑔(𝜃)𝜒 . The
382
chemical sourc function, 𝑓(𝜃) is more likely to be symmetric with latitude for Jupiter
383
(obliquity is almost zero), for example, 𝑓(𝜃)~(cos 𝜃)𝑛 .
384
385
4.2.1 Pure Advection Cases
386
387
The general approach to solve the pure advection 2-D cases is discussed in Appendix A.
388
We introduced two typical circulation patterns: (1) an axis-symmetric equator-to-pole
13
389
circulation pattern in each hemisphere, called 𝜓𝐴 , and (2) a pole-to-pole circulation
390
pattern in the entire hemisphere, called 𝜓𝐵 . The solutions are summarized in Table 2.
391
392
We now test our numerical model against analytical solutions. The numerical model has
393
dimensions 80×33, with 80 pressure grid points from 100 mbar to 5 mbar, and 33
394
latitudes from 85°S to 85°N with increments of 5°. Two numerical schemes are tested. In
395
the first scheme, called the “normal 2-D” mode, the photochemistry, diffusion, and
396
advection are solved together using a time-marching method (Shia et al. 1990). In the
397
second scheme, called the “quasi 2-D” mode (Liang et al. 2005), first we perform a series
398
of 1-D chemical-diffusive calculations at different latitudes using the matrix inversion
399
method (see Section 3), and then the meridional advection and horizontal diffusion are
400
applied to connect different latitudes. Our calculations show that, when reaching the
401
steady state, the two modes converge in the same solution. But the “quasi 2-D” mode
402
takes shorter time to reach the steady state than the “normal 2-D” mode, because the
403
former allows very large time step in the 1-D diffusion calculation but the latter is limited
404
by the CFL criterion for every time step.
405
406
We assume the Jupiter value 𝑁0 = 4.83 × 1018 𝑔 𝑐𝑚−3 at 100 mbar, and planetary
407
radius 𝑎 = 7.1824 × 109 𝑐𝑚 . We assume 𝑃0 = 10−16 𝑐𝑚−3 𝑠 −1 , 𝑤0 = 10−3 𝑐𝑚 𝑠 −1 ,
408
𝐻 = 2.5 × 105 𝑐𝑚, and 𝜆 = 0.3. So the transport timescale is about 3 × 109 𝑠. For each
409
circulation pattern, we tested two fictitious chemical tracers, the short-lived tracer with
410
9
loss rate faster than transport (𝐿−1
0 = 10 𝑠) and the long-lived tracer with loss rate
411
11
slower than transport (𝐿−1
𝑠).
0 = 10
412
413
In the case of Jupiter with nearly no obliquity, we hypothesize the circulation pattern to
414
be axis-symmetric (𝜓𝐴 ), in which air rises at the equator and sinks at the poles, our
415
analytical solution is given in case IX, with 𝐺(𝑒 −𝜉 𝜓) = 5 × 10−19 (𝑒 −𝜉 𝜓) . For the
416
other circulation pattern, which might be relevant to the planets with large obliquity, such
417
as Earth or Titan, air rises at the south pole and sinks at the north pole, and our analytical
418
solution is given in case X, with 𝐺(𝑒 −𝜉 𝜓) = −0.26(𝑒 −𝜉 𝜓)
2
14
−1
. We use the boundary
419
values from the analytical solutions for the numerical model. The mass stream functions
420
(scaled by the air density for each layer) are shown in Figs. 8 and 9, respectively.
421
422
The results from case IX are plotted in Fig. 8, with the numerical simulation results (dots)
423
on top of each curve. For the short-lived tracer, chemical production and loss dominates
424
its local abundances. Because the production is higher at equator and loss is higher at the
425
poles, the steady-state mixing ratio of the short-lived tracer is higher in the low latitudes
426
and lower in the high latitudes in the upper pressure levels (~5-10 mbar), while in the
427
lower pressure levels (~50-100 mbar), its latitudinal distribution is also affected by the
428
lower boundary conditions. On the other hand, although with the production and loss rate,
429
the latitudinal distribution of the long-lived species has the opposite trend as the short-
430
lived species, with higher mixing ratio in the higher latitudes and lower mixing ratio in
431
the lower latitudes in the upper pressure levels, showing the transport dominant results. In
432
the lower pressure levels, the solutions are also affected by the lower boundary
433
conditions. The numerical results are based on our 2-D CTM (Caltech/JPL Kinetics
434
Model), which is able to reproduce the analytical results almost perfectly. The largest
435
differences between the analytical and numerical simulations are found in two poles, but
436
still less than 4%. Case IX qualitatively interprets the opposite latitudinal distributions
437
between C2H2 (a short-live species) and C2H6 (a long-lived species), as revealed by the
438
Cassini and Voyager spectra (Nixon et al. 2010; Zhang et al. 2012a).
439
440
Similar behavior is revealed by the results from case X (Fig. 9). Although both tracers
441
have higher production rate in the southern hemisphere and linear loss rate coefficient
442
independent of latitude, their steady-state latitudinal distributions are significantly
443
different. Again, the short-lived tracer is dominated by the chemistry, while the long-
444
lived tracer shows transport-dominant behavior.
445
446
4.2.3 Pure Diffusive Cases
447
448
Ignoring the advection terms, we now have a 2-D diffusion equation
15
449
1
𝜕
𝜕𝜒
𝑒𝜉 𝜕
𝜕𝜒
(cos 𝜃 𝐾𝑦𝑦 ) + 2 (𝐾0 𝑒 (𝛾−1)𝜉 ) + 𝑃0 𝑒 (𝛼+1)𝜉 𝑓(𝜃) − 𝐿0 𝑒 𝛽𝜉 𝑔(𝜃)𝜒
2
𝑎 cos 𝜃 𝜕𝜃
𝜕𝜃
𝐻 𝜕𝜉
𝜕𝜉
450
= 0.
(13)
451
452
Generally there is no analytical solution for this equation. If we further assume the
453
solution of the 2-D diffusion equation can be expressed as: 𝜒(𝜃, 𝜉) = 𝐴(𝜃)𝐺(𝜉), and let
454
𝐺(𝜉) = 𝐶1 𝑒 (1−𝛾)𝜉 , 𝛼 = −𝛾, and 𝛽 = 0, the solutions for the pure diffusion 2-D cases are
455
discussed in Appendix B. Four typical cases are summarized in Table 3. In the numerical
456
11
simulations, we assume 𝑃0 = 10−16 𝑐𝑚−3 𝑠 −1 , 𝐿−1
0 = 10 𝑠 , 𝛾 = 0.1 . The results are
457
plotted in Fig. 10.
458
459
In case XI, we assume the production rate is uniform at the top and a linear loss rate.
460
𝐶1 = 0 in the solution leads to a trivial case with a constant mixing ratio with latitude. If
461
𝐶1 ≠ 0, the solution will end up with a horizontal diffusional equilibrium solution. We
462
assume 𝐾𝑦𝑦 = 2.1 × 109 𝑐𝑚2 𝑠 −1 in the numerical simulations. The solution shows a
463
bowl-shaped distribution with a minimum value at the equator and maximum value at the
464
poles (Fig. 10). So the horizontal flux is from pole to equator. At any latitude there
465
should be a flux convergence to balance the chemical loss. And the area weighted loss
466
rate has a maximum at equator, therefore a maximum flux convergence as well.
467
468
If the production rate is not uniform at the top and there is no chemical loss, the analytical
469
solutions for case XII (𝑛 = 0) and case XIII (𝑛 = 1) exist. In the numerical simulations,
470
we assume 𝐾𝑦𝑦 = 1 × 109 𝑐𝑚2 𝑠 −1 . The solutions show an upside-down bowl-shaped
471
distribution with the maximum value at the equator and a sharp falloff in the polar region
472
(Fig. 10). So the horizontal flux is from equator to pole. Since there is no chemical loss in
473
these cases, at any latitude there should be a flux divergence to balance the chemical
474
production. The solutions demonstrate that, if the chemical loss can be ignored, any
475
diffusive transport process with a chemical production rate that is either flat (𝑛 = 0) or
476
peak (𝑛 = 1) in the low latitudes will result in a high mixing ratio in the low latitudes.
477
Therefore, the horizontal mixing solution of C2H6, whose chemical loss can be ignored,
478
will have an upside-down bowl shape. Our simple analytical cases are consistent with the
16
479
model results in Liang et al. (2005) and Lellouch et al. (2006), but contrary to the
480
Voyager and Cassini data (Zhang et al. 2012a). We note that the ratio of production rate
481
with the horizontal mixing determines the ‘flatness’ of the bowl-shape distribution. A
482
more efficient horizontal mixing leads to a flatter distribution. The horizontal timescale
483
should be much shorter than the chemical production timescale in the Liang et al. (2005)
484
case, which is correct because in the lower atmosphere chemical production of C 2H6 is
485
not efficient. The observed latitudinal distribution from CIRS implies either the mean
486
residue circulation plays an important role, or there could be a chemical source of C2H6 in
487
the polar regions, although there is no evidence for any possible ion chemistry initiated
488
by precipitating particles in the aurora region that could enhance the ethane abundances.
489
But this possible chemical mechanism should not significantly enhance the abundances
490
of C2H2, which, in principle, is much more sensitive to the local chemical source but the
491
observations do not show any enhancement of C2H2 in the polar region.
492
493
However, the solution (case XIV) also shows a bowl-shaped distribution with the
494
maximum value at the equator and minimum value at the poles (Fig. 10). So if we ignore
495
the advection terms, the required latitudinal slope of the production rate of C2H6 should
496
be much steeper than the sin2 𝜃 in order to explain the CIRS observations. Numerical
497
simulations with more realistic chemistry and eddy mixing are needed to confirm this
498
hypothesis.
499
500
5. 2-D SYSTEM IN THE ZONAL PLANE
501
502
If we consider the system in an altitude-longitude plane, which is appropriate for slowly
503
rotating planets such as Venus and hot Jupiters, a strong subsolar-anti-solar circulation
504
coupled with the zonal jets, and with the inhomogeneous production rate, will lead to a
505
different scenario. In the zonal plane, the horizontal eddy diffusion can be neglected. In
506
the steady state, the governing equation is
507
𝑢 𝜕𝜒
𝜕𝜒
𝜕
𝜕𝜒
𝑃−𝐿
+𝑤
− 𝑒 𝜉 (𝑒 −𝜉 𝐾𝑧𝑧 ) =
.
𝑎 𝜕𝜆
𝜕𝑧
𝜕𝑧
𝜕𝑧
𝑁
17
(14)
508
where 𝜆 is longitude, 𝑎 is radius of the latitude circle. We can formulate the
509
streamfunction 𝜓 in the longitude and vertical plane:
𝜕 −𝜉
(𝑒 𝜓),
𝜕𝑧
1 𝜕𝜓
𝑤=
.
𝑎 𝜕𝜆
510
𝑢 = −𝑒 𝜉
511
(15𝑎)
(15𝑏)
512
If we neglect the diffusion term, this is basically the same as the 2-D problem in the
513
meridional plane, but without the cos 𝜃 factor. The solution is related to the subsolar-anti-
514
solar circulation, which is analogous to that in the equator-pole circulation problem (case
515
IX).
516
517
We can also simply solve the problem by neglecting the vertical advection term in
518
equation (14). Suppose a fast jets rapidly flows along the latitude circle with a constant
519
velocity 𝑢, the air mass is approximately conserved in that altitude over a short timescale,
520
such as the four-day zonal wind on Venus and fast zonal jets on hot Jupiters. The vertical
521
profile is modified by the eddy diffusion. As usual, we assume 𝑃(𝜆, 𝜉) = 𝑃0 𝑁0 𝑒 𝛼𝜉 𝑓(𝜆),
522
𝐿(𝜆, 𝜉) = 𝐿0 𝑁0 𝑒 (𝛽−1)𝜉 𝑔(𝜆)𝜒, and 𝐾𝑧𝑧 = 𝐾0 𝑒 𝛾𝜉 . Equation (14) becomes
523
𝑢 𝜕𝜒
𝜕
𝜕𝜒
− 𝑒 𝜉 (𝐾0 𝑒 (𝛾−1)𝜉 ) − 𝑃0 𝑒 (𝛼+1)𝜉 𝑓(𝜆) + 𝐿0 𝑒 𝛽𝜉 𝑔(𝜆)𝜒 = 0.
𝑎 𝜕𝜆
𝜕𝑧
𝜕𝑧
524
Let us check a simple case with a special solution. Similar to the argument of the 2-D
525
diffusive system, we let 𝛼 = −𝛾 and 𝛽 = 0, we assume the solution can be expressed as:
526
𝜒(𝜆, 𝜉) = 𝐴(𝜆)𝐺(𝜉), where 𝐺(𝜉) = 𝐶1 𝑒 (1−𝛾)𝜉 . The solution is
527
𝜒(𝜆, 𝜉) = e−
𝑎𝐿0
𝑔(𝜆)𝑑𝜆
𝑢 ∫
[𝐶1 +
𝑎𝐿0
𝑎𝑃0
∫ 𝑓(𝜆)e 𝑢 ∫ 𝑔(𝜆)𝑑𝜆 𝑑𝜆] 𝑒 (1−𝛾)𝜉 .
𝑢
(16)
(17)
528
The periodic boundary condition, i.e., 𝜒(0, 𝜉) = 𝜒(2𝜋, 𝜉), requires 𝐶1 to be 0. A simple
529
case would be 𝑓(𝜆) = 1 + cos 𝑘𝜆, where 𝑘 = 1 stands for the two-modal production rate
530
(day-night contrast). The solution would be
531
𝜒(𝜆, 𝜉) =
𝑃0
1
[1 +
cos(𝑘𝜆 − 𝜙)]𝑒 (1−𝛾)𝜉 .
2
𝐿0
√1 + 𝑞
(18)
532
where the dimensionless variable 𝑞 = 𝑘𝑢/𝑎𝐿0 measures chemical loss timescale versus
533
the advection timescale (across an envelope of the production rate distribution). 𝜙 =
534
tan−1 𝑞 can be regarded as the phase lag of the mixing ratio distribution compared with
18
535
the production rate distribution and the amplitude of the mixing ratio variation is smaller
536
than the production rate variation by a factor of √1 + 𝑞 2 . When the advection timescale
537
and the chemical loss timescale are comparable, i.e., 𝑞 = 1, the phase shift is 45°. If 𝑞 is
538
large, i.e., when the chemical loss is slower than the advection timescale, the zonal wind
539
will quickly redistribute the chemicals and lead to a large phase lag and smooth the
540
mixing ratio profile in longitude. On the other hand, if 𝑞 is small, chemistry will
541
dominate the distribution. Therefore the phase lag due to advection would be smaller
542
since the local chemical equilibrium will be established more quickly, leading to a large
543
mixing ratio bulge along the latitude circle. Fig. 11 illustrates some typical results at 𝜉 =
544
0 for a range of values of 𝑞.
545
546
6. CONCLUDING REMARKS
547
548
In this study we systematically investigated the possible analytical benchmark cases in
549
the chemical-advective-diffusive system. Although our solutions are highly idealized, we
550
can still gain physical insights on what control the vertical and latitudinal profiles of the
551
short-lived and long-lived species in the stratosphere of Jupiter. In the 1-D system, we
552
show that CH4 and C2H6 are mainly in diffusive equilibrium, and C2H2 profile can be
553
approximated by the modified Bessel functions. Those analytical solutions could be used
554
for the simple treatment of photochemistry in climate models or general circulation
555
models. In the 2-D system in the meridional plane, analytical solutions for two typical
556
circulation patterns are derived. Simple tracer transport cases demonstrate that the
557
distribution of short-lived species is dominated by the local chemical sources and sinks,
558
while that of the long-lived species is significantly influenced by the circulation. This
559
may help solve the difference in the latitudinal distribution between C2H2 and C2H6, as
560
revealed by the Cassini and Voyager spectra. On the other hand, it seems difficult for a
561
pure diffusive transport process to produce a similar latitudinal profile of C2H6 whose
562
chemical loss can be neglected. Intuitively it also makes sense because the horizontal
563
eddy mixing is not able to reverse the latitudinal gradient driven by the photochemistry.
564
Therefore, unless there is a missing chemical source for C2H6 (but not for C2H2) in the
565
polar region, the most probable solution is a meridional circulation from equator to pole.
19
566
The detailed structure of the residue circulation in the stratosphere of Jupiter requires a
567
realistic numerical simulation. For the slowly rotating planet, which might have
568
longitudinal heterogeneous chemical sources, the interaction between the advection by
569
the zonal wind and chemistry might cause a phase lag between the final tracer
570
distribution and the original source distribution. The magnitude of the phase lag and
571
longitudinal contrast of the tracer profile depends on the relative strength between the
572
advection timescale and the lifetime of the tracer. This is similar to the mechanism that
573
causes a phase shift between the locations of the atmospheric temperature maximum and
574
sub-solar point (where heating is a maximum) on close-in giant planets (Knutson et al.
575
2007).
576
577
The analytical solutions have been used to validate the numerical simulations from our 2-
578
D Caltech/JPL CTM and show good agreement for various cases. The largest discrepancy
579
usually happens in the polar region, especially when the analytical solutions have
580
singular values at the poles, such as case XI. Increasing the horizontal and vertical
581
resolution would lead to better agreement. This study lays the theoretical basis and
582
numerical tools for future realistic chemistry-transport modeling in planetary and
583
exoplanetary atmospheres.
584
585
586
587
588
589
590
591
592
593
594
595
596
20
597
598
FIGURE CAPTIONS
599
600
Fig. 1. Simulated hydrocarbon profiles from the idealized model, the C2 chemistry model
601
(with realistic eddy and molecular diffusivities as the full chemistry model) and the full
602
chemistry model.
603
604
Fig. 2. Profiles of eddy diffusivity and molecular diffusivity (for CH4) in the idealized
605
model (eddy profile I) compared with the full chemistry models (eddy profile II). The
606
blue curve is the total diffusivity for the idealized model.
607
608
Fig. 3. CH4 from numerical simulations compared with analytical solutions. The dashed
609
lines are asymptotic profiles.
610
611
Fig. 4. C2H6 from numerical simulations compared with analytical solutions. The dashed
612
lines are asymptotic profiles.
613
614
Fig. 5. C2H2 from numerical simulations compared with analytical solutions. The dashed
615
lines are asymptotic profiles.
616
617
Fig. 6. Analytical CH4 profiles from the cases with and without wind. 𝑤0 =
618
±5 × 10−8 𝑐𝑚 𝑠 −1 .
619
620
Fig. 7. Analytical C2H6 profiles from the cases with and without wind. 𝑤0 =
621
±1 × 10−6 𝑐𝑚 𝑠 −1 . The dotted line is with the molecular diffusion in the upper
622
atmosphere.
623
624
Fig. 8. Plots of case IX. Upper panel: Analytic mass stream functions in units of
625
𝑔 𝑐𝑚−1 𝑠 −1 . Bottom panel: comparison of analytical (lines) and numerical (dots)
626
solutions for two fictitious chemical tracers, the short-lived tracer (orange) and the long-
21
627
lived tracer (blue). The solid and dashed curves correspond to 5 and 50 mbar,
628
respectively.
629
630
Fig. 9. Plots of case X. Upper panel: Analytic mass stream functions in units of
631
𝑔 𝑐𝑚−1 𝑠 −1 . Bottom panel: comparison of analytical (lines) and numerical (dots)
632
solutions for two fictitious chemical tracers, the short-lived tracer (orange) and the long-
633
lived tracer (blue). The solid and dashed curves correspond to 5 and 50 mbar,
634
respectively.
635
636
Fig. 10. Comparison of analytical (lines) and numerical (dots) solutions for the pure
637
diffusive 2-D cases XI, XII, XIII, and XIV in Table 3. The case number is indicated in
638
each panel. The solid and dashed curves correspond to 5 and 50 mbar, respectively.
639
640
Fig. 11. Analytical solutions at 𝜉 = 0 for the zonal transport cases. Different ratios (𝑞
641
values) of the transport versus chemical timescales result in different phase lags and
642
different amplitudes of the longitudinal mixing ratio profiles. The dashed lines indicate
643
the longitudes corresponding to the peaks of the mixing ratio profiles. The chemical
644
source distribution follows a cosine function with its peak at longitude 0°.
645
646
647
648
649
650
651
652
653
654
655
656
657
22
658
659
660
661
REFERENCES
662
663
Allen, M., Yung, Y.L., & Waters, J.W., 1981, J. Geophys. Res. 86: 3617.
664
Andrews, D.G., Holton, J.R., & Leovy, C.B., 1987, Middle atmosphere dynamics.
665
666
(Academic Pr).
Chamberlain, T.P., & Hunten, D.M., 1989, Theory of planetary atmospheres: an
667
introduction to their physics and chemistry. (Academic Press).
668
Conrath, B.J., Gierasch, P.J., & Leroy, S.S., 1990, Icarus 83(2): 255.
669
Friedson, A.J., West, R.A., Hronek, A.K., Larsen, N.A., & Dalal, N., 1999, Icarus 138(1):
670
141.
671
Gladstone, G.R., Allen, M., & Yung, Y., 1996, Icarus 119(1): 1.
672
Knutson, H.A., Charbonneau, D., Allen, L.E., et al., 2007, Nature 447(7141): 183.
673
Lellouch, E., Bézard, B., Strobel, D., et al., 2006, Icarus 184(2): 478.
674
Liang, M.C., Shia, R.L., Lee, A.Y.T., et al., 2005, ApJ 635(2): L177.
675
Lin, J.S., & Hildemann, L.M., 1997, Atmos. Environ. 31(1): 59.
676
Lindzen, R.S., 1981, J. Geophys. Res.-Oceans and Atmos. 86, 9707.
677
Moses, J., Fouchet, T., Bézard, B., et al., 2005, J. Geophys. Res. 110(E8): E08001.
678
Nixon, C.A., Achterberg, R.K., Conrath, B.J., et al., 2007, Icarus 188(1): 47.
679
Nixon, C.A., Achterberg, R.K., Romani, P.N., et al., 2010, Planet. Space Sci. 58(13):
680
681
682
1667.
Polyanin, A.D., & Zaitsev, V.F., 2002, Handbook of exact solutions for ordinary
differential equations. (Chapman & Hall/CRC).
683
Prather, M.J., 1986, J. Geophys. Res 91(6): 6671.
684
Shia, R., Ha, Y.L., Wen, J.S., & Yung, Y.L., 1990, J. Geophys. Res. 95(D6): 7467.
685
Strobel, D., 1974, ApJ 192: L47.
686
West, R., Friedson, A., & Appleby, J., 1992, Icarus 100(1): 245.
687
Yelle, R.V., Griffith, C.A., & Young, L.A., 2001, Icarus 152(2): 331.
688
Zhang, X., Nixon, C.A., Shia, R.L., et al., 2012a, Planet. Space Sci., submitted.
23
689
Zhang, X., West. R., Banfield, D., et al., 2012b, Planet. Space Sci., to be submitted.
690
691
692
693
694
Fig. 1.
695
696
697
698
699
700
24
701
702
703
Fig. 2.
704
705
706
707
25
708
709
710
Fig. 3.
711
26
712
713
714
Fig. 4.
715
716
717
27
718
719
720
Fig. 5.
721
722
723
724
28
725
726
Fig. 6.
727
728
729
29
730
731
732
Fig. 7.
733
734
735
736
737
738
30
739
740
Fig. 8.
741
742
743
31
744
745
746
Fig. 9.
747
748
749
750
32
751
752
753
Fig. 10.
754
755
756
757
758
759
760
761
33
762
763
764
Fig. 11.
765
766
34