446 METR Test # 3 November 13, 2001
Name: Solution
1) (25 pts) Assume the ocean is a gray-body with a wavelength-independent absorptance of 0.85. If the temperature of the ocean is 20 oC, at what solar zenith angle will the absorbed solar irradiance be equal to the emitted irradiance? You can neglect absorption or reflection within the atmosphere.
Eout = T 4 = aT 4 Ein = aEo cos Eout = Ein aT 4 = aEo cos cos =
-1
T 4
T 4 = cos -1 E Eo o
(5.67 10 -8 mWK 4 )(293 K ) 4 2 = 75.0 o = cos W 1370 m 2
2) (30 pts) If the absorption coefficient of the gases in the atmosphere (assume all have the same average value) is 3.0x10-5 m2/kg, how much total solar radiant energy is absorbed by the atmosphere every second? You can neglect reflectance, and can assume that the absorption coefficient is independent of wavelength.
u = k c dx =
0
x
TOA
k dz = k dz = k dz
c c a 0 0 0 po
TOA
TOA
From hydrostatic equation (and HW #5) (101300 Pa )(3.0 10 pk so, u = u = o = 9.81 sm g 2 a = 1 - e -u = 0.267
dp = - po = - a gdz = - g
0
0
TOA
TOA
dz
a 0
-5 m 2 kg
)
N m2
kg m s2
kg N
= 0.310
2 Energy absorbed = a ( Earth cross sec tional area) Eo = aREarth Eo / W = (0.267)(6.38 106 m) 2 (1370 m 2 )( JW s ) = 4.68 1016 J s
3) (15 pts) A transmissiometer measures the decrease in monochromatic radiance (from a laser) over a long path length. Sometimes these devices consist of a laser and a detector separated by a long distance, while other times they consist of a laser and detector side-by-side, with a placed mirror some distance away that reflects the radiation coming from the laser back to the detector. If 72% of the radiation were transmitted if the detector were located at point B below, how much would be transmitted if instead a mirror were located at point B, and the detector back at point A?
B (detector or mirror)
A (laser and sometimes detector)
Ground
=
L = e -u Lo
x
u AB = k c dx = - ln A B = - ln(0.72) = 0.329
0
u AB A = 2 k c dx = 2u AB = 0.657
0
x
A B A
=e
-u AB A
= 0.518
4) (5 pts) Is the wavelength of maximum radiance of a gray-body (a = constant, but not 1) greater than, less than, or equal to that of a black-body (a = 1) at the same temperature?
If the absorptance does not vary with wavelength, the entire monochromatic radiance curve will be shifted downward since emissivity is equal to absorptance (Kirchhoff's Law), but its shape will remain the same. So the wavelength of maxim...