in equation (5) is given by convergence of net radiative heat flux which is calculated by using radiative transfer equation. We consider following radiation processes in this model; absorption of near infrared solar radiation (NIR), absorption and emission of infrared radiation associated with atmospheric CO, absorption and scattering of solar radiation, and absorption and emission of infrared radiation associated with dust.
is represented as follows.
(22) |
Both infrared and near infrared radiative flux associated with
CO are calculated by Goody narrow band
model (c.f., Goody and Young, 1989).
In calculating infrared radiative flux, CO
15 m band is only considered.
The upward and downward infrared radiative flux
and the infrared radiative heating rate per unit mass
are calculated as follows.
is line strength, is square root of the product of line strength and line width and is the reference value of , is effective path length, and is reference pressure (= 1013 hPa).
In calculating near infrared solar radiative flux,
CO 4.3 m, 2.7 m, and 2.0 m band are considered.
The near infrared solar radiative flux
and are calculated as follows.
where
, is the solar
zenith angle, and is the solar radiative flux
per unit wave length at the top of atmosphere which is represented
as follows.
(29) | |||
(30) |
where is the surface temperature of the sun (= 5760 K), is the Stefan-Boltzmann constant (= 5.67 WmK), is solar constant on the mean radius of Mars orbit (= 591 Wm), and is the radius of Mars orbit and its mean value, is solar radiative flux at the top of atmosphere. is depend on season, latitude and local time. Detail descriptions of and are shown in 第5.3節.
The transmission function averaged over
in near
infrared wavelength region is similar to that in infrared
wavelength region except for the effective path length .
The number of narrow band and its band width are similar to those of Savijärvi (1991a). The line strength and the square root of the product of line strength and line width are quoted from those at 220 K listed by Houghton (1986). These vaues are listed in Table 4 Table 7.
CO 15 m band ranges from 500 cm to 900 cm and 4.3 m band ranges from 2200 cm to 2450 cm, where is equal to 25 cm. CO 2.7 m band ranges from 3150 cm to 4100 cm and 4.0 m band ranges from 4600 cm to 5400 cm , where is equal to 100 cm.
(cm) | (cm) | ||||
512.5 | 1.952 | 2.870 | 712.5 | 1.232 | 8.387 |
537.5 | 2.785 | 1.215 | 737.5 | 2.042 | 2.852 |
562.5 | 5.495 | 2.404 | 762.5 | 7.278 | 6.239 |
587.5 | 5.331 | 1.958 | 787.5 | 1.337 | 2.765 |
612.5 | 5.196 | 5.804 | 812.5 | 3.974 | 8.897 |
637.5 | 7.778 | 2.084 | 837.5 | 1.280 | 3.198 |
662.5 | 8.746 | 7.594 | 862.5 | 2.501 | 1.506 |
687.5 | 2.600 | 2.635 | 887.5 | 3.937 | 1.446 |
(cm) | (cm) | ||||
2212.5 | 9.504 | 2.866 | 2337.5 | 5.587 | 1.206 |
2237.5 | 2.217 | 3.000 | 2362.5 | 6.819 | 1.182 |
2262.5 | 4.566 | 1.134 | 2387.5 | 1.256 | 8.873 |
2287.5 | 7.965 | 2.011 | 2412.5 | 7.065 | 3.404 |
2312.5 | 1.055 | 5.880 | 2437.5 | 8.522 | 4.236 |
(cm) | (cm) | ||||
3150 | 1.324 | 9.836 | 3650 | 1.543 | 3.245 |
3250 | 7.731 | 4.900 | 3750 | 1.649 | 2.722 |
3350 | 1.232 | 2.952 | 3850 | 1.180 | 9.535 |
3450 | 5.159 | 7.639 | 3950 | 1.464 | 2.601 |
3550 | 4.299 | 1.914 | 4050 | 1.251 | 2.021 |
(cm) | (cm) | ||||
4650 | 2.185 | 1.916 | 5050 | 8.778 | 2.012 |
4750 | 2.040 | 6.475 | 5150 | 8.346 | 1.804 |
4850 | 1.197 | 3.112 | 5250 | 8.518 | 8.474 |
4950 | 4.829 | 5.759 | 5350 | 4.951 | 1.597 |
The solar and infrared radiative flux associated with dust are calculated by using the -Eddington approximation (c.f., Liou, 1980). The -Eddington approximation is well used in calculating radiative transfer with anisotropic scattering. The asymmetry factor of dust for solar and infrared radiation are between 0 and 1 which means forward scattering occurs.
The upward and downward diffuse solar radiative flux per unit wave
length associated with dust
,
are obtained as solutions of following
equations.
The boundary condition of
(31) and
(32)
are that
at the top
of atmosphere and
at the surface, where is the surface albedo.
are expressed as follows.
where
are optical depth, single scattering
albedo and asymmetry factor scaled by -Eddington
approximation, which are given as follows.
where are optical depth, single scattering albedo and asymmetry factor, respectively.
The upward and downward infrared radiative flux per unit wave length
associated with dust are obtained as solutions of similar equations
used for calculation of diffuse solar flux
((31), (32))
except for the
last term in right hand side of each equation.
The boundary condition of
(33)
and (34)
is that
at the top
of atmosphere and
is equal to
at the surface.
The Plank function
in
(33)
and (34) is averaged over the band width.
are the lower and upper wave length of the band.
The radiative heating rate associated with dust is calculated as follows.
is the direct solar radiative flux per unit wave length,
(37) |
The dust opacity is calculated by using the mass mixing ratio
and effective radius of dust.
In this model, we suppose that the size distribution of dust particle
is the modified gamma distribution (Toon et al., 1977).
The monoclomatic optical depth is represented
by using the extinction coefficient per unit volume
as follows.
(41) |
(42) | |||
(43) |
(44) |
The extinction efficiency is defined as
the ration of extinction cross section to geometric cross section.
(45) |
Similarly,
the scattering efficiency and
absorption efficiency is defined as follows.
(46) | |||
(47) |
In this model, the dust opacity is derived from the
mass mixing ratio of atmospheric dust.
Given parameters are the cross section weighted mean extinction
efficiency
, the single scattering albedo
, the size distribution function of dust
, the mode radius , the effective (or, cross section
weighted mean) radius , and the density of dust
particle .
and are defined as
follows, respectively.
(48) | |||
(49) |
Supposing that the shape of scattering particle is sphere,
the extinction coefficient per unit mass is given as follows.
(50) |
The values of band width and optical parameters of dust (extinction efficiency, single scattering albedo, asymmetry factor) considered in this model are following to those of Forget et al. (1999) except for 11.6-20 m band of dust. The overlap between visible band of dust and CO near infrared band is omitted.
The 5-11.6 m infrared dust opacity is obtained by dividing the visible dust opacity by the visible to infrared opacity ratio , which is set to be 2. (Forget, 1998). The 20-200 m infrared dust opacity is calculated by using and the value of shown in Table 8.
Band(m) | Band(cm) | |||
0.1-5 m | 2000-10 | 1.0 | 0.920 | 0.55 |
5-11.6 m | 870-2000 | 0.253 | 0.470 | 0.528 |
20-200 m | 50-500 | 0.166 | 0.370 | 0.362 |
Parameters | Standard values | Note |
3.04 | Ockert-Bell, et al. (1997) | |
2 | Forget (1998) | |
2.5 m | Pollack et al. (1979) | |
0.4 m | Pollack et al. (1979) |
The solar flux at the top of atmosphere is depend on season, latitude and local time. In this section, we show as a function of local time at a specified season and latitude.
Suppose that (Wm) is solar constant on the mean orbital radius
of Planet, and is the radius of
orbit and its mean value, is solar zenith
angle, is latitude, is the solar
inclination, is the hour angle
(
, is length of day ).
is represented by using these variables as follows.
(52) | |||
(53) |
(54) |
Parameters | Standard values | Note |
20N | Pollack et al. (1979) | |
100 | 〃 | |
0.093 | ||
25.2 | ||
110 | Carr (1996), Fig. 1 | |
591 Wm |