1
Appendix A
2
Analytical Solutions For Atmospheric Radiative Equilibrium State
3
4
A.1 Infrared (IR) Source Function in Radiative Equilibrium
5
6
The general equation of radiative transfer is:
7
𝜇
𝑑𝐼
=𝐼−𝐵
𝑑𝜏
(𝐴. 1)
8
where 𝐼 is the IR radiation field (intensity), 𝐵 is the atmospheric source function, 𝜏 is the
9
optical depth and 𝜇 is the cosine of the view angle. Under Eddington approximation:
10
𝐼(𝜇) = 𝐼0 + 𝐼1 𝜇, the moments are:
11
𝐼 ̅ = ∫ 𝐼(𝜇) 𝑑𝜇 𝑑𝜔 = 𝐼0
12
𝐹 = ∫ 𝜇𝐼 𝑑𝜇 𝑑𝜔 =
4𝜋
𝐼
3 1
(𝐴. 2𝑏)
13
𝐾 = ∫ 𝜇 2 𝐼 𝑑𝜇 𝑑𝜔 =
4𝜋
𝐼
3 0
(𝐴. 2𝑐)
14
(𝐴. 2𝑎)
Substituting into Equation (A.1) and we obtain:
15
4𝜋
𝑑𝐵
𝑑2𝐹
= 3𝐹 − 2
𝑑𝜏
𝑑𝜏
(𝐴. 3)
16
The upper boundary condition of thermal source function 𝐵(𝜏) can be estimated from the
17
two-stream approximation. Suppose we only have one upward stream 𝐼 + , and one
18
downward stream 𝐼 − :
19
𝐹 = 𝜋(𝐼 + − 𝐼 − )
𝐼̅ =
20
21
22
23
24
25
𝐼+ + 𝐼−
2
(𝐴. 4𝑎)
(𝐴. 4𝑏)
So Equation (A. 3) becomes:
𝑑𝐹(𝜏)
= 2𝐹(𝜏) + 4𝜋𝐼 − (𝜏) − 4𝜋𝐵(𝜏)
𝑑𝜏
(𝐴. 5)
If we assume no downward IR flux (𝐼 − ) at the upper boundary, we obtain:
𝐵(0) =
𝐹(0) 1 𝑑𝐹
−
( )
2𝜋
4𝜋 𝑑𝜏 𝜏=0
The radiative equilibrium state requires:
1
(𝐴. 6)
26
𝑑𝐹
𝑑𝐹ℎ𝑒𝑎𝑡
=−
= −𝑆(𝜏)
𝑑𝜏
𝑑𝜏
27
where 𝐹(𝜏) is the thermal IR flux emitted by the atmosphere and 𝐹ℎ𝑒𝑎𝑡 is the heat flux
28
from outside heat source. 𝑆(𝜏) = 𝑑𝐹ℎ𝑒𝑎𝑡 /𝑑𝜏 is called the heating rate. The primary heat
29
flux is from the downward solar flux 𝐹𝑠 , which is absorbed in the visible or near IR
30
wavelengths so that it does not exchange the photons with the thermal IR wavelengths.
31
On the other hand, there could be an upward heat flux 𝐹⨁ , either from (1) the surface that
32
is heated by absorbing the sunlight directly; or (2) the radioactive decay of the heavy
33
elements in the crust of terrestrial plants; or (3) the upward convective heat flux from the
34
deep interior of giant planets. 𝐹⨁ is mainly in the thermal IR region and usually treated
35
as the lower boundary condition of the upward flux.
(𝐴. 7)
36
37
Here we discuss two cases:
38
(a) For a certain incident angle, solar Flux at 𝜏𝑣 is:
39
𝐹𝑠− (𝜇, 𝜏𝑣 ) = −𝜇𝐹𝑠 𝑒
𝜏
− 𝑣
(𝐴. 8)
𝜇
40
where 𝜏𝑣 is the optical depth at visible (or near IR) wavelength 𝑣 where the solar flux can
41
be absorbed. 𝜇 is the cosine of the solar incident angle 𝜇 = 𝑐𝑜𝑠𝜃, 𝐹𝑠 is the solar flux at
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the top of the atmosphere (TOA). Negative sign means downward flux. The radiative
43
equilibrium thermal flux 𝐹𝐼𝑅 = −𝐹ℎ𝑒𝑎𝑡 = 𝐹⨁ + 𝜇𝐹𝑠 𝑒
44
depth in the thermal emission wavelengths and 𝛼 = 𝜏𝜈 /𝜏. From Equation (A. 3), we solve
45
the 𝐵(𝜏):
46
47
𝐵(𝜏) =
−
𝛼𝜏
𝜇
, where 𝜏 is the averaged optical
3𝐹⨁ 2
3𝜇𝐹𝑠 2 𝜇
𝛼 𝜇 −𝛼𝜏
( + 𝜏) +
[ + + ( − )𝑒 𝜇 ]
4𝜋 3
4𝜋 3 𝛼
3𝜇 𝛼
(𝐴. 9)
This is basically the same as the result in Goody and Yung (1995).
48
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(b) For a global averaged solar flux (including the diurnally average):
𝜋
∫02 𝐹𝑣− (𝜇, 𝜏𝑣 ) 2𝜋𝑅 2 𝑠𝑖𝑛𝜃𝑑𝜃
1
−
̅̅̅̅
𝐹 (𝜏𝑣 ) =
=
−
𝐹 𝐸 (𝜏 )
4𝜋𝑅 2
2 𝑠 3 𝑣
50
∞ 𝑒 −𝑥𝑡
∞ 𝑥 𝑛−1 𝑒 −𝑡
(𝐴. 10)
51
where 𝐸𝑛 (𝑥) = ∫1
52
this integral should be understood in terms of the Cauchy principal value, due to the
𝑡𝑛
𝑑𝑡 = ∫𝑥
𝑡𝑛
𝑑𝑡 is the exponential integral, if 𝑥 is negative,
2
53
singularity in the integrand at zero. When 𝜏𝑠 = 0 (TOA), ̅̅̅̅
𝐹 − (𝜏𝑣 ) = 𝐹𝑠 /4. The radiative
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equilibrium thermal flux 𝐹𝐼𝑅 = −𝐹ℎ𝑒𝑎𝑡 = 𝐹⨁ + 2 𝐹𝑠 𝐸3 (𝛼𝜏). From Equation (A. 2), we
55
solve the 𝐵(𝜏):
56
1
𝐵(𝜏) =
3𝐹⨁ 2
3𝐹𝑠 1 + 𝛼 𝛼
1
( + 𝜏) +
[
+ 𝐸2 (𝛼𝜏) −
𝐸 (𝛼𝜏)]
4𝜋 3
4𝜋 6𝛼
6
2𝛼 4
(𝐴. 11)
57
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A. 2. Analytical Solutions For Radiance, Flux, Heating and Cooling Rate
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Based on the radiative equilibrium IR source functions above, we derive the analytical
61
solutions for the TOA radiance, upward and downward fluxes for each layer, and solar
62
heating and thermal cooling rates for each layer. The source function 𝐵(𝜏) could have
63
one of the following forms, corresponding to the above cases (a) and (b), respectively:
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1. 𝐵(𝜏) = 𝐵0 + 𝐵1 𝜏 + 𝐵2 𝑒 −𝛼𝜏
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2. 𝐵(𝜏) = 𝐵0 + 𝐵1 𝜏 + 𝐵2 𝐸2 (𝛼𝜏) + 𝐵3 𝐸4 (𝛼𝜏)
66
We will focus on the form 1 because it is less complicated than the form 2 and therefore
67
the analytical solutions can be obtained through the integrations. Form 1 can describe the
68
typical tropospheric temperature and stratospheric temperature with or without inversion.
69
In the following notation, we will ignore the wavenumber index in the lower subscript
70
but all the formulae should be considered monochromatic and can be used for testing the
71
numerical line-by-line (LBL) radiative transfer models. Please see Appendix D for the
72
details of the integration derivations.
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A.2.1. TOA Radiance
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The IR upward radiance at emission angle 𝜃 at 𝜏𝑖 :
𝐼 + (𝜏𝑖 , 𝜇) = 𝐵(𝜏𝑠 )𝑒
𝜏 −𝜏
− 𝑠 𝑖
𝜇
𝜏𝑠
+ ∫ 𝐵(𝑡)𝑒
𝜏𝑖
−
𝑡−𝜏𝑖
𝜇
𝑑𝑡
𝜇
(𝐴. 12)
78
where 𝜏𝑠 is the total optical depth from the upper to the lower boundary. 𝜇 = 𝑐𝑜𝑠𝜃.
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IR downward radiance at 𝜏𝑖 :
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𝐼
− (𝜏
𝜏𝑖
𝑖 , 𝜇)
= ∫ 𝐵(𝑡)𝑒
0
3
𝑡−𝜏𝑖
𝜇
𝑑𝑡
𝜇
(𝐴. 13)
81
We are more interested the TOA IR radiance (𝜏𝑖 = 0) that can be measured by the
82
satellite. The TOA IR radiance at emission angle 𝜃:
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𝐼
+ (0,
𝜇) = 𝐵(𝜏𝑠
𝜏
− 𝑠
)𝑒 𝜇
𝜏𝑠
+ ∫ 𝐵(𝑡)𝑒
−
0
84
𝑡
𝜇
𝑑𝑡
𝜇
(𝐴. 14)
The global averaged TOA IR radiance:
𝜋
𝜏𝑠
∫02 𝐼 + (0, 𝜇) 2𝜋𝑅 2 𝑠𝑖𝑛𝜃𝑑𝜃
+
̅̅̅
𝐼 (0, 𝜇) =
= 𝐵(𝜏𝑠 )𝐸2 (𝜏𝑠 ) + ∫ 𝐵(𝑡)𝐸1 (𝑡)𝑑𝑡 (𝐴. 15)
2𝜋𝑅 2
0
85
86
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Plug in the source function 𝐵(𝜏) = 𝐵0 + 𝐵1 𝜏 + 𝐵2 𝑒 −𝛼𝜏 , we obtain:
𝐼
+ (0,
𝜇) = (𝐵0 + 𝐵1 𝜏𝑠 + 𝐵2 𝑒
−𝛼𝜏𝑠
𝜏
− 𝑠
)𝑒 𝜇
𝜏𝑠
+ ∫ (𝐵0 + 𝐵1 𝑡 + 𝐵2 𝑒 −𝛼𝑡 )𝑒
−
𝑡
𝜇
0
88
= 𝐵0 + 𝐵1 𝜇(1 − 𝑒
𝜏
− 𝑠
𝜇
𝐵
2
) + 𝛼𝜇+1
(1 + 𝛼𝜇𝑒
1
𝜇
−(𝛼+ )𝜏𝑠
𝑑𝑡
𝜇
)
(𝐴. 16)
89
and
90
̅̅̅
𝐼 + (0, 𝜇) = (𝐵0 + 𝐵1 𝜏𝑠 + 𝐵2 𝑒 −𝛼𝜏𝑠 )𝐸2 (𝜏𝑠 ) + ∫ (𝐵0 + 𝐵1 𝑡 + 𝐵2 𝑒 −𝛼𝑡 )𝐸1 (𝑡)𝑑𝑡
𝜏𝑠
0
91
92
1
𝐵2
= 𝐵0 + 𝐵1 ( − 𝐸3 (𝜏𝑠 )) + [ln(1 + 𝛼) − 𝑒 −𝛼𝜏𝑠 𝐸1 (𝜏𝑠 ) + 𝐸1 ((1 + 𝛼)𝜏𝑠 )
2
𝛼
−𝛼𝜏𝑠
+ 𝛼𝑒
𝐸2 (𝜏𝑠 )]
(𝐴. 17)
93
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A.2.2. Upward and Downward Fluxes
95
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From the downward solar flux in Eqn. (A.8), the global-averaged solar flux at 𝜏𝑣 is:
𝜋
∫02 𝐹𝑣− (𝜇, 𝜏𝑣 ) 2𝜋𝑅 2 𝑠𝑖𝑛𝜃𝑑𝜃
1
−
̅̅̅̅
𝐹 (𝜏𝑣 ) =
= − 𝐹𝑠 𝐸3 (𝜏𝑣 )
2
4𝜋𝑅
2
97
(𝐴. 18)
98
From Eqn. (A.12), the IR upward flux at layer 𝜏𝑖 is
99
𝐹 + (𝜏𝑖 ) = ∫ 𝜇𝐼 + (𝜏𝑖 , 𝜇) 𝑑𝜇𝑑𝜔 = 2𝜋𝐵(𝜏𝑠 )𝐸3 (𝜏𝑠 − 𝜏𝑖 ) + 2𝜋 ∫ 𝐵(𝑡)𝐸2 (𝑡 − 𝜏𝑖 )𝑑𝑡 (𝐴. 19)
1
𝜏𝑠
0
100
𝜏𝑖
From Eqn. (A.13), the IR downward flux at layer 𝜏𝑖 :
1
101
𝐹
− (𝜏
𝑖)
= ∫ 𝜇𝐼
− (𝜏
𝜏𝑖
𝑖 , 𝜇) 𝑑𝜇𝑑𝜔
0
102
= 2𝜋 ∫ 𝐵(𝑡)𝐸2 (𝜏𝑖 − 𝑡)𝑑𝑡
0
Plug in the source function 𝐵(𝜏) = 𝐵0 + 𝐵1 𝜏 + 𝐵2 𝑒 −𝛼𝜏 , we obtain:
4
(𝐴. 20)
103
104
105
𝐹 + (𝜏𝑖 )
1 𝜏𝑖
= 𝜋𝐵0 + 2𝜋𝐵1 ( + − 𝐸4 (𝜏𝑠 − 𝜏𝑖 ))
3 2
−𝛼𝜏𝑖
2𝜋𝐵2 𝑒
+
{𝑒 −𝛼(𝜏𝑠 −𝜏𝑖 ) [(𝐸1 (𝜏𝑠 − 𝜏𝜏 ) − 𝛼𝐸2 (𝜏𝑠 − 𝜏𝑖 ) + 𝛼 2 𝐸3 (𝜏𝑠 − 𝜏𝑖 )]
𝛼2
106
− 𝐸1 ((1 + 𝛼)(𝜏𝑠 − 𝜏𝑖 )) − ln(1 + 𝛼) + 𝛼}
107
and
108
𝐹 − (𝜏𝑖 )
109
110
(𝐴. 21)
1
1 𝜏𝑖
= 2𝜋𝐵0 ( − 𝐸3 (𝜏𝑖 )) + 2𝜋𝐵1 (− + + 𝐸4 (𝜏𝑖 ))
2
3 2
−𝛼𝜏𝑖
2𝜋𝐵2 𝑒
+
{𝑒 𝛼𝜏𝑖 [(𝐸1 (𝜏𝑖 ) + 𝛼𝐸2 (𝜏𝑖 )] − 𝐸1 ((1 − 𝛼)𝜏𝑖 ) − ln(|𝛼 − 1|) − 𝛼}
2
𝛼
(𝐴. 22)
111
112
A.2.3. Solar Heating Rate
113
114
The Solar heating Rate at 𝜏𝑣 is:
115
116
𝐻ℎ𝑒𝑎𝑡 (𝜏𝑣 ) =
𝜏
𝑑𝐹𝑠− (𝜇, 𝜏𝑣 )
− 𝑣
= 𝐹𝑠 𝑒 𝜇
𝑑𝜏
(𝐴. 23)
And the global-averaged solar heating rate at 𝜏𝑣 :
− (𝜏 )
̅̅̅̅
𝑑𝐹
1
𝑣
= 𝐹𝑠 𝐸2 (𝜏𝑣 )
𝑑𝜏
2
117
̅̅̅̅̅̅̅
𝐻ℎ𝑒𝑎𝑡 (𝜏𝑣 ) =
118
But the zonally averaged solar heating rate at 𝜏𝑣 is not integrable.
(𝐴. 24)
119
120
A.2.4. Thermal Cooling Rate
121
122
123
124
125
The IR Cooling Rate at 𝜏𝑖 :
𝐻𝑐𝑜𝑜𝑙 =
𝑑𝐹(𝜏𝑖 )
= 2𝜋[−𝐵(𝜏𝑖 )𝐸2 (𝜏𝑖 ) + (𝐵(𝜏𝑠 )−𝐵(𝜏𝑖 ))𝐸2 (𝜏𝑠 − 𝜏𝑖 ) + 𝐼1 + 𝐼2 ]
𝑑𝜏
where
𝜏𝑠
𝐼1 = ∫ (𝐵(𝑡) − 𝐵(𝜏𝑖 ))𝐸1 (𝑡 − 𝜏𝑖 )𝑑𝑡
𝜏𝑖
126
𝜏𝑖
𝐼2 = ∫ (𝐵(𝑡) − 𝐵(𝜏𝑖 ))𝐸1 (𝜏𝑖 − 𝑡)𝑑𝑡
0
5
(𝐴. 25)
127
Plug in the source function 𝐵(𝜏) = 𝐵0 + 𝐵1 𝜏 + 𝐵2 𝑒 −𝛼𝜏 , we obtain:
128
𝐻𝑐𝑜𝑜𝑙 =
129
= −2𝜋𝐵0 𝐸2 (𝜏𝑖 ) + 2𝜋𝐵1 (𝐸3 (𝜏𝑖 ) − 𝐸3 (𝜏𝑠 − 𝜏𝑖 ))
130
+
131
− 𝐸1 ((1 − 𝛼)𝜏𝑖 )−𝑒 −𝛼(𝜏𝑠 −𝜏𝑖 ) 𝐸1 (𝜏𝑠 − 𝜏𝑖 ) + 𝐸1 ((1 + 𝛼)(𝜏𝑠 − 𝜏𝑖 ))
132
+ ln(1 + 𝛼) − ln(|𝛼 − 1|) − 2𝛼}
𝑑𝐹(𝜏𝑖 )
𝑑𝜏
2𝜋𝐵2 𝑒 −𝛼𝜏𝑖
{𝛼𝑒 −𝛼(𝜏𝑠 −𝜏𝑖 ) 𝐸2 (𝜏𝑠 − 𝜏𝑖 ) + 𝑒 𝛼𝜏 𝐸1 (𝜏𝑖 )
𝛼
(𝐴. 26)
133
134
135
136
137
138
139
140
141
142
143
144
145
146
147
148
149
150
151
152
153
154
155
6
156
Appendix B
157
Numerical Schemes for Flux and Cooling Rate Calculations
158
159
We consider two numerical schemes for the calculations of upward and downward
160
intensities, fluxes and cooling rate. The first method is the direct integral with Gaussian
161
Quadrature method. But usually it needs larger number of Gaussian points to achieve the
162
satisfactory accuracy. Our second method is the finite difference scheme. In the following
163
notation, we ignore the wavenumber index in the lower subscript but all the formulae
164
should be considered monochromatic and line-by-line (LBL) radiative transfer
165
calculations. Using optical depth as the vertical coordinate, We divide the atmosphere
166
into 𝑁 layers with 𝑁 + 1 atmospheric levels as layer boundaries, noted by 𝜏𝑖 , where 𝑖 =
167
1, … , 𝑁 + 1. The source function 𝐵𝑖 = 𝐵(𝜏𝑖 ) is defined on the levels. Within each layer
168
[𝜏𝑖 , 𝜏𝑖+1 ], we approximate the source function linearly as function of optical depth:
169
𝐵𝑖 (𝑡) = 𝐵𝑖 + 𝐵𝑖′ (𝑡 − 𝜏𝑖 ) , where 𝜏𝑖 ≤ 𝑡 ≤ 𝜏𝑖+1 , where 𝐵𝑖′ = (𝐵(𝜏𝑖+1 ) − 𝜏(𝜏𝑖 ))/∆𝑖 and
170
∆𝑖 = 𝜏𝑖+1 − 𝜏𝑖 .
171
172
The upward intensity 𝑰+
𝒗 (𝝉𝒊 , 𝝁):
𝑁
173
𝐼
+ (𝜏
𝑖 , 𝜇) = 𝐵𝑁+1 𝑒
−(𝜏𝑁+1 −𝜏𝑖 )/𝜇
𝜏𝑗+1
+ ∑∫
𝐵𝑗 (𝑡)𝑒 −(𝑡−𝜏𝑖 )/𝜇
𝑗=1 𝜏𝑗
𝑑𝑡
𝜇
𝑁
174
= 𝐵𝑁+1 𝑒
−(𝜏𝑁+1 −𝜏𝑖 )/𝜇
+ ∑ 𝑒 −(𝜏𝑗 −𝜏𝑖 )/𝜇 [(𝐵𝑗 + 𝐵𝑗′ 𝜇)(1 − 𝑒 −∆𝑗/𝜇 ) − 𝐵𝑗′ ∆𝑗𝑒 −∆𝑗/𝜇 ]
(𝐵. 1)
𝑗=1
175
176
Specifically, the TOA emission is:
𝑁
177
𝐼 + (0, 𝜇) = 𝐵𝑠 𝑒 −𝜏𝑠 /𝜇 + ∑ 𝑒 −𝜏𝑗/𝜇 [(𝐵𝑗 + 𝐵𝑗′ 𝜇)(1 − 𝑒 −∆𝑗/𝜇 ) − 𝐵𝑗′ ∆𝑗𝑒 −∆𝑗/𝜇 ]
(𝐵. 2)
𝑗=1
178
179
The downward intensity 𝑰−
𝒗 (𝝉𝒊 , 𝝁):
𝑖−1
180
𝐼 − (𝜏𝑖 , 𝜇) = ∑ 𝑒 −(𝜏𝑗−𝜏𝑖 )/𝜇 [𝐵𝑗 (𝑒 ∆𝑗/𝜇 − 1) + 𝐵𝑗′ 𝜇(
𝑗=1
7
∆𝑗 ∆𝜏/𝜇
𝑒
− 𝑒 ∆𝑗/𝜇 + 1)]
𝜇
(𝐵. 3)
181
182
The upward flux 𝑭↑ (𝝉𝒊 ):
183
↑ (𝜏
𝑁
𝐹
𝑖)
= 2𝜋𝐵𝑁+1 𝐸3 (𝜏𝑁+1 − 𝜏𝑖 ) + 2𝜋 ∑ 𝐼𝑖𝑗↑
(𝐵. 4)
𝑗=𝑖
184
where 𝐼𝑖𝑗↑ = 𝐵𝑗 [𝐸3 (𝜏𝑗 − 𝜏𝑖 ) − 𝐸3 (𝜏𝑗+1 − 𝜏𝑖 )] − 𝐵𝑗′ [(𝜏𝑗+1 − 𝜏𝑗 )𝐸3 (𝜏𝑗+1 − 𝜏𝑗 ) −
185
𝐸4 (𝜏𝑗 − 𝜏𝑖 ) + 𝐸4 (𝜏𝑗+1 − 𝜏𝑖 )]
186
187
If ∆𝑗 < 10−4 , we use an approximated formula to minimize the numerical errors:
188
𝐼𝑖𝑗↑ =
3𝐵𝑗 − 𝐵𝑗+1
𝐸2 (𝜏𝑗+1 − 𝜏𝑖 )∆𝑗
2
189
190
The downward flux 𝑭↓ (𝝉𝒊 ):
𝑖−1
191
𝐹
↓ (𝜏
𝑖)
= 2𝜋 ∑ 𝐼𝑖𝑗↓
(𝐵. 4)
𝑗=1
192
where 𝐼𝑖𝑗↓ = 𝐵𝑗 [𝐸3 (𝜏𝑗 − 𝜏𝑗+1 ) − 𝐸3 (𝜏𝑖 − 𝜏𝑗 )] + 𝐵𝑗′ [(𝜏𝑗+1 − 𝜏𝑗 )𝐸3 (𝜏𝑖 − 𝜏𝑗+1 ) +
193
𝐸4 (𝜏𝑖 − 𝜏𝑗 ) − 𝐸4 (𝜏𝑖 − 𝜏𝑗+1 )]
194
195
If ∆𝑗 < 10−4 , we use:
196
𝐼𝑖𝑗↓ =
𝐵𝑗 + 𝐵𝑗+1
𝐸2 (𝜏𝑖 − 𝜏𝑗+1 )∆𝑗
2
197
198
The cooling rate can be calculated in two ways.
199
200
201
(1) The cooling rate defined at level 𝜏𝑖 is:
𝐻𝑐𝑜𝑜𝑙
𝑁
𝑖−1
𝑗=𝑖
𝑗=1
𝑑𝐹(𝜏𝑖 )
=
= 2𝜋[𝐵𝑁+1 𝐸2 (𝜏𝑁+1 − 𝜏𝑖 ) − 2𝐵𝑖 + ∑ 𝐼1 + ∑ 𝐼2 ]
𝑑𝜏
202
where 𝐼1 = 𝐵𝑗 [𝐸2 (𝜏𝑗 − 𝜏𝑖 ) − 𝐸2 (𝜏𝑗+1 − 𝜏𝑖 )] − 𝐵𝑗′ [(𝜏𝑗+1 − 𝜏𝑗 )𝐸2 (𝜏𝑗+1 − 𝜏𝑗 ) −
203
𝐸3 (𝜏𝑗 − 𝜏𝑖 ) + 𝐸3 (𝜏𝑗+1 − 𝜏𝑖 )]
204
205
If ∆𝑗 < 10−4 (and 𝑗 ≠ 𝑖 − 1), we use:
8
(𝐵. 5)
206
𝐼1 =
3𝐵𝑗 − 𝐵𝑗+1
𝐸1 (𝜏𝑗+1 − 𝜏𝑖 )∆𝑗
2
207
208
and 𝐼2 = 𝐵𝑗 [𝐸2 (𝜏𝑗 − 𝜏𝑗+1 ) − 𝐸2 (𝜏𝑖 − 𝜏𝑗 )] + 𝐵𝑗′ [(𝜏𝑗+1 − 𝜏𝑗 )𝐸2 (𝜏𝑖 − 𝜏𝑗+1 ) +
209
𝐸3 (𝜏𝑖 − 𝜏𝑗 ) − 𝐸3 (𝜏𝑖 − 𝜏𝑗+1 )]
210
211
If ∆𝑗 < 10−4 (and 𝑗 ≠ 𝑖 − 1), we use:
212
𝐼2 =
𝐵𝑗 + 𝐵𝑗+1
𝐸1 (𝜏𝑖 − 𝜏𝑗+1 )∆𝑗
2
213
214
215
(2) The averaged cooling rate within layer [𝜏𝑖 , 𝜏𝑖+1 ] is:
𝐻𝑐𝑜𝑜𝑙 =
𝑑𝐹(𝜏𝑖 )
= [𝐹 ↓ (𝜏𝑖 ) + 𝐹 ↑ (𝜏𝑖+1 ) − 𝐹 ↓ (𝜏𝑖+1 ) − 𝐹 ↑ (𝜏𝑖 )]/(𝜏𝑖+1 − 𝜏𝑖 )
𝑑𝜏
(𝐵. 6)
216
217
Comparison Between Analytical and Numerical Calculations
218
219
In order to test the accuracy of the two numerical schemes, we applied three typical
220
source function profiles to the models:
221
222
(1) 𝐵(𝜏) = 3 + 0.2 𝜏
223
(2) 𝐵(𝜏) = 5 + 0.05 𝜏 + 5 exp(−0.5 𝜏)
224
(3) 𝐵(𝜏) = 1 + 0.01 𝜏 + 15 exp(−0.9 𝜏)
225
226
For the Gaussian Quadrature method we use 100 Gaussian points. The model has 72
227
levels with optical depth evenly distributed logarithmically from 10-5 to 102. Fig. B1
228
shows the three source function profiles and the corresponding averaged IR cooling rates
229
within each layer, indicated by the middle point of the vertical grid. The comparisons
230
with analytical solutions show that the two integration methods are roughly comparably
231
accurate and the relative errors compared with the analytical solutions are generally less
232
than 1 percent. The differences are larger in the larger optical depth region because (1)
233
the cooling rate is approaching zero, thus limited by the computational precision; (2) the
234
optical grid is coarser, leading to larger numerical errors. For instance, the 720-level
9
235
model agrees much better than the 72-level model. The cooling rate for case (1) has the
236
least errors because the source function is linear in 𝜏, in consistent with the assumption in
237
our numerical methods within each grid box. When the non-linear terms are more
238
significant, the relative errors become larger. Fig. B2 shows the averaged IR cooling rates
239
defined at levels and the associated errors. Fig. B3 and B4 are showing the upward and
240
downward thermal fluxes at each level, respectively. The relative difference of the
241
numerical values and analytical solutions are generally within 1 percent.
242
243
244
245
246
247
248
249
250
251
252
253
254
255
256
257
258
259
260
261
262
263
264
265
10
266
Appendix C
267
Non-LTE Formulism
268
269
In this study we consider a simple two-level non-LTE radiative transfer between a
270
vibrational level and the ground level. We ignore the interaction between the vibrational
271
levels, especially the excitation of the low-lying fundamentals (e.g., in the thermal IR
272
wavelength) by the higher vibrational levels (i.e., the vibrational-vibration transfer, or
273
“V-V transfer”). In the absence of the vibrational chemistry, collisions will be the only
274
cause of the departure of the source function from LTE through the vibrational-
275
translational transfer (“V-T transfer”). The non-LTE source function, 𝐽ν at wavenumber
276
ν, can be formulated as (see Appleby, 1990; Yelle et al., 1991; Goody and Yung, 1995;
277
López-Puertas and Taylor, 2001):
278
𝐽ν (𝜏) =
𝐼 (̅ 𝜏) + 𝜀(𝜏)𝐵ν (𝜏)
1 + 𝜀(𝜏)
(𝐶. 1)
279
where ε ≡ C10 /A10 , C10 is the de-activation rates due to thermal collisions and A10 is the
280
Einstein coefficient of the vibrational band. 𝐵ν is the LTE source function (Planck
281
function). 𝐼 ̅ is the mean radiation field averaged in the vibrational band:
𝐼 (̅ 𝜏) =
282
∫ 𝐼ν̅ (𝜏)𝑘ν 𝑑ν
∫ 𝑘ν 𝑑ν
(𝐶. 2)
283
where 𝐼ν̅ (𝜏) is the mean intensity averaged for all the emission angles (Goody and Yung,
284
2.103):
285
𝐼ν̅ (𝜏) =
1
1 𝜏𝑠
𝐵ν (𝜏𝑠 )𝐸2 (𝜏𝑠 − 𝜏) + ∫ 𝐽ν (𝑡)𝐸1 (|𝑡 − 𝜏|)𝑑𝑡
2
2 0
(𝐶. 3)
286
where 𝜏 is the optical depth. 𝐸1 (𝑥) and 𝐸2 (𝑥) are exponential integral functions. For gas
287
giant planets, we need to include the continuum effect from H2-H2 and H2-He CIA that is
288
assumed to be always in LTE. The source function is modified to include this
289
partitioning:
290
𝐽ν (𝜏) =
1 − 𝑟(𝜏)
[𝐼 (̅ 𝜏) + ε(𝜏)𝐵ν (𝜏)] + 𝑟(𝜏)𝐵ν (𝜏)
1 + ε(𝜏)
(𝐶. 4)
291
where 𝑟(𝜏) = 𝜏𝐻2 −𝐻2 /(𝜏𝑔𝑎𝑠 + 𝜏𝐻2 −𝐻2 ) is the ratio of the continuum optical depth to the
292
total optical depth.
11
293
Substituted in Eqn. (C.3) and we obtain the equation for the mean radiation field:
294
𝐼ν̅ (𝜏)
295
=
296
1 𝜏𝑠 1 − 𝑟(𝑡) ∫ 𝐼ν̅ (𝑡)𝑘ν 𝑑ν
+ ∫ {
[
+ ε(t)𝐵ν (𝑡)] + 𝑟(𝑡)𝐵ν (𝑡)} 𝐸1 (|𝑡
2 0 1 + ε(t)
∫ 𝑘ν 𝑑ν
297
− 𝜏|)𝑑𝑡
298
Once we solve 𝐼ν̅ (𝜏), and the source function 𝐽ν (𝜏) can be derived. We now introduce
299
two numerical schemes to solve 𝐼ν̅ (𝜏).
1
𝐵 (𝜏 )𝐸 (𝜏 − 𝜏)
2 ν 𝑠 2 𝑠
(𝐶. 5)
300
301
C.1 Iteration Method
302
303
In the iteration method, first we write the equations in the discretized format:
𝐽ν (𝜏) =
304
1 − 𝑟(𝜏) ∑ 𝐼ν̅ (𝜏)𝑘ν Δν
[
+ ε(𝜏)𝐵ν (𝜏)] + 𝑟(𝜏)𝐵ν (𝜏)
∑ 𝑘ν Δν
1 + ε(𝜏)
(𝐶. 6𝑎)
𝑁
𝐽ν (𝜏𝑗 ) + 𝐽ν (𝜏𝑗+1 )
1
1
𝐼̅ν (𝜏) = 𝐵ν (𝜏𝑁 )𝐸2 (𝜏𝑁 − 𝜏) + ∑
|𝐸2 (|𝜏 − 𝜏𝑗 |) − 𝐸2 (|𝜏 − 𝜏𝑗+1 |)|
2
2
2
{
𝑗=1
305
Start from 𝐼̅ν (𝜏) = ∫ 𝐵ν (𝜏)𝑘ν 𝑑ν / ∫ 𝑘ν 𝑑ν and iterate the above two following equations.
306
Usually 𝐼̅ν (𝜏) converges after four or five iterations.
307
308
C.2 Matrix Inversion Method
309
310
We can also directly solve 𝐼̅ν (𝜏) by the Matrix Inversion Method. Discretize the equation:
311
𝐼ν̅ (𝜏𝑖 )
312
1
1
= 𝐵ν (𝜏𝑁 )𝐸2 (𝜏𝑁 − 𝜏𝑖 ) + ∑[(1 − 𝑟𝑗 )𝐽𝑔𝑗 + 𝑟𝑗 𝐽𝑐𝑗 ]|𝐸2 (|𝜏𝑖 − 𝜏𝑗 |) − 𝐸2 (|𝜏𝑖 − 𝜏𝑗+1 |)|
2
2
𝑁
𝑗=1
313
314
315
316
(𝐶. 7)
where 𝐽𝑔𝑗 and 𝐽𝑐𝑗 are the continuum and gas band contribution:
𝐽𝑐𝑗 =
(𝐵𝑗 + 𝐵𝑗+1 )
1 𝐼𝑗̅ + εj 𝐵𝑗 𝐼̅̅̅̅̅
𝑗+1 + εj+1 𝐵𝑗+1
& 𝐽𝑔𝑗 = [
+
]
2
2 1 + εj
1 + εj+1
Therefore:
12
(𝐶. 8)
(𝐶. 6𝑏)
317
𝐼ν̅ (𝜏𝑖 )
318
𝑟𝑗 + εj
𝑟𝑗 + εj+1
1
1
= 𝐵𝑁 𝐸2 (𝜏𝑁 − 𝜏𝑖 ) + ∑[
𝐵𝑗 +
𝐵 ] |𝐸2 (|𝜏𝑖 − 𝜏𝑗 |) − 𝐸2 (|𝜏𝑖 − 𝜏𝑗+1 )||
2
4
1 + εj
1 + εj+1 𝑗+1
𝑁
𝑗=1
319
𝑁
(1 − 𝑟𝑗 )𝐼̅𝑗 (1 − 𝑟𝑗 )𝐼̅̅̅̅̅
1
𝑗+1
+ ∑(|𝐸2 (|𝜏𝑖 − 𝜏𝑗 | − 𝐸2 (|𝜏𝑖 − 𝜏𝑗+1 |)|)[
+
]
4
1 + εj
1 + εj+1
(𝐶. 9)
𝑗=1
320
Using Einstein notation, 𝐼ν̅ (𝜏𝑖 ) ≡ 𝛼𝑖 + 𝛽𝑖𝑘 𝐼𝑘̅ , where:
321
𝛼𝑖
322
=
323
𝑟𝑗 + εj
𝑟𝑗 + εj+1
1
+ ∑(
𝐵𝑗 +
𝐵 )|𝐸 (|𝜏 − 𝜏𝑗 |) − 𝐸2 (|𝜏𝑖 − 𝜏𝑗+1 |)|
4
1 + εj
1 + εj+1 𝑗+1 2 𝑖
1
𝐵 (𝜏 )𝐸 (𝜏 − 𝜏𝑖 )
2 ν 𝑁 2 𝑁
𝑁
𝑗=1
324
1
1 − 𝑟1
𝛽𝑖,1 = (|𝐸2 (|𝜏𝑖 − 𝜏1 |) − 𝐸2 (|𝜏𝑖 − 𝜏2 |)|)(
)
4
1 + ε1
325
𝛽𝑖,𝑘 (𝑘 = 2 … 𝑁)
326
=
327
1
1 − 𝑟𝑘
+ |𝐸2 (|𝜏𝑖 − 𝜏𝑘 |) − 𝐸2 (|𝐸𝑖 − 𝜏𝑘+1 |)|(
)
4
1 + εk
328
1
1 − 𝑟𝑁
𝛽𝑖,𝑁+1 = (|𝐸2 (|𝜏𝑖 − 𝜏𝑁 |) − 𝐸2 (|𝜏𝑖 − 𝜏𝑁+1 |)|)(
)
4
1 + εN+1
329
Take the band average (“~” stands for the matrix form):
330
1
1 − 𝑟𝑘−1
|𝐸2 (|𝜏𝑖 − 𝜏𝑘−1 |) − 𝐸2 (|𝜏𝑖 − 𝜏𝑘 |)|(
)
4
1 + εk
∫ 𝛼̃ 𝑘ν 𝑑ν ∫ 𝛽̃ 𝑘ν 𝑑ν
̃ + 𝛽̃̅ 𝐼 ̅
𝐼̃̅ =
+
𝐼 ̅ = 𝛼̅
∫ 𝑘ν 𝑑ν
∫ 𝑘ν 𝑑ν
−1
(𝐶. 10)
331
Therefore the final solution is 𝐼̃̅ = (𝑰̃ − 𝛽̃̅ )
332
̃.
LTE situation (ε → ∞), 𝛽̃̅ → 0̃, 𝐼̃̅ → 𝛼̅
333
Our calculations show that the matrix inversion method is less accurate than the iteration
334
method. Therefore in this study we adopt the latter.
335
336
337
13
𝛼̃
̅ , where 𝑰̃ is the identity matrix. For the
338
Appendix D
339
Derivations for the Integral Involving Exponential Integral Functions
340
341
The general properties of the exponential integral functions are:
1
(𝑛 ≥ 2)
𝑛−1
342
𝐸𝑛 (0) =
343
∫ 𝐸𝑛 (𝑥)𝑑𝑥 = −𝐸𝑛+1 (𝑥)
344
Therefore, we obtain:
𝜏
345
𝜏
∫ 𝑡𝐸𝑛 (𝑡)𝑑𝑡 = −𝜏𝐸𝑛+1 (𝜏) + ∫ 𝐸𝑛+1 (𝑡)𝑑𝑡 =
0
0
1
− 𝐸𝑛+2 (𝜏) − 𝜏𝐸𝑛+1 (𝜏)
𝑛+1
346
𝝉
347
D.1. Derivation of 𝑭𝒏 = ∫𝝉 𝒔 𝒆−𝜶𝒕 𝑬𝒏 (𝒕 − 𝝉)𝒅𝒕 :
348
𝐹𝑛 = 𝑒 −𝛼𝜏 ∫
𝜏𝑠 −𝜏
𝑒 −𝛼𝑡 𝐸𝑛 (𝑡)𝑑𝑡
0
𝑒
−𝛼𝜏
349
=−
350
Therefore,
[𝑒 −𝛼(𝜏𝑠 −𝜏) 𝐸𝑛 (𝜏𝑠 − 𝜏) −
𝛼
1
+ 𝑒 𝛼𝜏 𝐹𝑛−1 ]
𝑛−1
𝑛
1 𝑛−1
1 𝑛−𝑚+1 −𝛼𝜏
𝑒 −𝛼𝜏
𝐹𝑛 = (− )
𝐹1 + ∑ (− )
[𝑒 𝑠 𝐸𝑚 (𝜏𝑠 − 𝜏) −
]
𝛼
𝛼
𝑚−1
351
(𝐷. 1)
𝑚=2
352
where,
353
𝐹1 = ∫ 𝑒 −𝛼𝑡 𝐸1 (𝑡 − 𝜏)𝑑𝑡 = 𝑒 −𝛼𝜏 ∫
𝜏𝑠
𝜏
𝜏𝑠 −𝜏
∞
𝑒 −𝛼𝑡 𝑑𝑡 ∫
0
1
𝑒 −𝑡𝑧
𝑑𝑧
𝑧
∞
1 − 𝑒 −(𝜏𝑠 −𝜏)𝑥
𝑒 −𝛼𝜏 ∞
1
1
𝑑𝑥 =
∫ (
− ) (1 − 𝑒 −(𝜏𝑠 −𝜏)𝑥 ) 𝑑𝑥
𝛼 1+𝛼 𝑥 − 𝛼 𝑥
1+𝛼 𝑥(𝑥 − 𝛼)
354
= 𝑒 −𝛼𝜏 ∫
355
=
356
𝑒 −𝛼𝜏
=
[−𝑒 −𝛼(𝜏𝑠 −𝜏) 𝐸1 (𝜏𝑠 − 𝜏) + 𝐸1 ((1 + 𝛼)(𝜏𝑠 − 𝜏)) + ln(1 + 𝛼)]
𝛼
𝑒 −𝛼𝜏
𝑒 −𝛼𝜏 ∞ 𝑒 −(𝜏𝑠 −𝜏)𝑥 𝑒 −(𝜏𝑠 −𝜏)𝑥
ln(1 + 𝛼) +
∫ (
−
)𝑑𝑥
𝛼
𝛼 1+𝛼
𝑥
𝑥−𝛼
357
358
when n=2,
14
(𝐷. 2)
359
𝜏𝑠
𝐹2 = ∫ 𝑒 −𝛼𝑡 𝐸2 (𝑡 − 𝜏)𝑑𝑡 =
𝜏
−𝛼𝜏
360
−
𝑒
{𝑒 −𝛼(𝜏𝑠 −𝜏) (𝛼𝐸2 (𝜏𝑠 − 𝜏) − 𝐸1 (𝜏𝑠 − 𝜏)) + 𝐸1 ((1 + 𝛼)(𝜏𝑠 − 𝜏)) + ln(1 + 𝛼) − 𝛼}
𝛼2
361
(𝐷. 3)
362
363
𝝉
D.2 Derivation of 𝑲𝒏 = ∫𝟎 𝒆−𝜶𝒕 𝑬𝒏 (𝝉 − 𝒕)𝒅𝒕 :
𝜏
364
1
𝑒 −𝛼𝜏
[𝐸𝑛 (𝜏) −
+ 𝐾𝑛−1 ]
𝛼
𝑛−1
𝐾𝑛 = 𝑒 −𝛼𝜏 ∫ 𝑒 𝛼𝑡 𝐸𝑛 (𝑡)𝑑𝑡 =
0
365
366
Therefore,
𝑛
𝐾1
1 𝑛−𝑚+1
𝑒 −𝛼𝜏
𝐾𝑛 = 𝑛−1 + ∑ ( )
[𝐸𝑚 (𝜏) −
]
𝛼
𝛼
𝑚−1
367
(𝐷. 4)
𝑚=2
368
where,
369
𝐾1 = ∫ 𝑒 −𝛼𝑡 𝐸1 (𝜏 − 𝑡)𝑑𝑡 = 𝑒 −𝛼𝜏 ∫ 𝑒 𝛼𝑡 𝑑𝑡 ∫
𝜏
𝜏
0
∞
0
∞
−𝜏𝑥
−𝛼𝜏
1
𝑒 −𝑡𝑧
𝑑𝑧
𝑧
∞
1−𝑒
𝑒
1
1
𝑑𝑥 =
∫ ( −
) (1 − 𝑒 −𝜏𝑥 ) 𝑑𝑥
𝑥(𝑥
+
𝛼)
𝛼
𝑥
𝑥
+
𝛼
1−𝛼
1−𝛼
370
= 𝑒 −𝛼𝜏 ∫
(𝐷. 5)
371
372
Note that if 𝛼 > 1, there is a singularity at 𝑥 = 1. But the Cauchy principal value still
373
exists. For example, if 𝛼 > 1,
∞
374
−𝜖
∞
1
1
1
∫
𝑑𝑥 = lim(∫
𝑑𝑥 + ∫
𝑑𝑥)
𝜖→0 1−𝛼 𝑥(𝑥 + 𝛼)
1−𝜏 𝑥(𝑥 + 𝛼)
𝜖 𝑥(𝑥 + 𝛼)
375
= lim
376
=−
𝛼+𝜖
ln (𝛼 − 𝜖 ) − ln(𝛼 − 1)
𝜖→0
𝛼
ln(𝛼 − 1)
𝛼
377
378
if 𝛼 < 1,
379
∫
∞
1
ln(1 − 𝛼)
𝑑𝑥 = −
𝛼
1−𝛼 𝑥(𝑥 + 𝛼)
380
15
381
Therefore,
382
−𝑒 −𝛼𝜏
𝑒 −𝛼𝜏 ∞ 𝑒 −𝜏𝑥 𝑒 −𝜏𝑥
𝐾1 =
ln(|𝛼 − 1|) +
∫ (
−
) 𝑑𝑥
𝛼
𝛼 1−𝛼 𝑥 + 𝛼
𝑥
383
=
𝑒 −𝛼𝜏 𝛼𝜏
[𝑒 𝐸1 (𝜏) − 𝐸1 ((1 − 𝛼)𝜏) − ln(|𝛼 − 1|)]
𝛼
(𝐷. 6)
384
385
when n=2,
386
𝐾2 = ∫ 𝑒 −𝛼𝑡 𝐸2 (𝜏 − 𝑡)𝑑𝑡
𝜏
0
387
𝑒 −𝛼𝜏 𝛼𝜏
= 2 {𝑒 [(𝐸1 (𝜏) + 𝛼𝐸2 (𝜏)] − 𝐸1 ((1 − 𝛼)𝜏) − ln(|𝛼 − 1|) − 𝛼}
𝛼
(𝐷. 7)
388
389
390
391
Note that there is no singularity at 𝛼 = 1, since:
−𝐸1 ((1 − 𝛼)𝜏) − ln(|𝛼 − 1|) = 𝛾 + ln(|𝛼 − 1|𝜏) − ln(|𝛼 − 1|) = 𝛾 + ln(𝜏)
where 𝛾 is the Euler–Mascheroni constant.
392
393
394
395
396
397
398
399
400
401
402
403
404
405
406
407
408
16
409
Figure Captions
410
411
Fig. B.1. Test cases (1: black; 2: orange; 3: blue) for averaged cooling rates within each
412
layer from different numerical schemes. Three symbols indicate three solutions. Squares
413
are analytical solutions; triangles are from the Gaussian Quadrature method; Crosses are
414
from the finite difference scheme. The relative difference is defined as the absolute value
415
of (numerical-analytical)/analytical.
416
417
Fig. B.2. Same as Fig. B.1, but for the cooling rates defined at levels.
418
419
Fig. B.3. Test cases of upward IR fluxes 𝐹 ↑ (𝜏) from different numerical schemes and
420
compared with the analytical solutions. The symbols and colors follow Fig. B.1.
421
422
Fig. B.4. Test cases of downward IR fluxes 𝐹 ↓ (𝜏) from different numerical schemes and
423
compared with the analytical solutions. The symbols and colors follow Fig. B.1.
17
424
425
Fig. B.1.
18
426
427
Fig. B.2.
428
429
430
431
432
19
433
434
Fig. B.3.
435
436
437
438
20
439
440
Fig. B.4.
441
442
21