San José State University

applet-magic.com
Thayer Watkins
Silicon Valley
& Tornado Alley
USA

Planetary Temperature and the Albedos
of Clouds and Surface for
Visible and Infrared Radiation

A planet's surface temperature is established by a balance of the incoming radiation which is absorbed and the outgoing radiation which escapes into space. Some radiation from the surface is captured by greenhouse gases in the atmosphere and they radiate in all directions, some of which reaches the planet's surface. But this is not the only mechanism by which energy radiated from the surface gets returned to the surface. Radiation can be reflected from the underside of clouds. This cloud blanket effect might compete with the greenhouse effect in raising a planet's surface temperature.

A previous study found that if the albedos for clouds and surface are the same for long wavelength (infrared) radiation as for short wavelength radiation (visible and ultraviolet) then the cloud cover of a planet does not raise its equilibrium temperature. The effect of the cloud cover to retain more of the long wavelength radiation is exactly offset by the clouds shielding the planet's surface from the solar short wavelength radiation.

This study is to examine the effect of planetary clouds on temperature if the albedos for surface and clouds for infrared radiation are different from those for visible and ultraviolet radiation.

The Energy Balance Model

A planet's surface absorbs some of the short wavelength (visible and ultraviolet) radiation impinging upon it and reflects some back out into space. The surface radiates long wavelength (infrared) radiation out into space based on its temperature.

Let αS be the proportion of short wavelength radiation reflected from a planet's surface. This is its albedo. Let σ be the solar constant, the intensity of the incoming solar radiation be unit area perpendicular to the radiation. The intensity of the outgoing long wavelength radiation is given kT4, where T is the absolute temperature of the surface in degrees Kelvin and k is a constant, known as the Stefan-Boltzmann constant and has a value of 5.67×10-8 Wm-2K-4. If R is the radius of the planet the net energy absorbed at the surface is σ(1−α)(πR²). Assuming the planet's surface has an average temperature T, the outgoing energy is kT4(4πR²). For balance then

4kT4(πR²) = σ(1−αS)(πR²)
which upon division of
both sides by the surface
area 4πR² gives
kT4 = (σ/4)(1−αS)
and hence
T = (σ(1−αS)/(4k))1/4

Thus the actual area of the planet is thus irrelevant and hereafter all computations will be per square meter of surface area.

Cloud Cover

Let βS be the albedo for short wavelength radiation of a cloud cover. Then a share of βS of the incoming radiation is reflected and a share of (1-βS) passes through the clouds. For an atmosphere transparent to the incoming radiation the energy incident upon the surface is σ(1-βS). A share (1-αS) is absorbed and σ(1-βS)α is reflected away from the surface. This impinges upon the underside of the clouds and a share of βS is reflected back down to the surface. The energy of this reflectance is σ(1-βSSβS. The next round of reflectances results in σ(1-βS)(αSβS)² arriving at the surface. At the next round it is σ(1-βS)(αSβS)³.

Altogether then the short wave energy arriving at the surface is

σ(1-βS) + σ(1-βSSβS + σ(1-βS)(αSβS)² + …)
= σ(1-βS)[1 + αSβS + (αSβS)² + …]
this is a geometric series
which evaluates to
σ(1-βS)/(1−αSβS)

The surface absorbs a share (1-αS) of this and thus the total energy absorbed by the surface from the short wavelength radiation is

σ(1-αS)(1-βS)/(1−αSβS)

On the other hand, a unit of surface radiates 4kT4 up to the underside of the clouds. If the albedos of the surface and clouds for long wavelength radiation are αL and βL, respectively, then the radiant of energy 4kT4βL comes back. A round of reflectances from the surface to the clouds and back again results in (4kT4βL)(αLβL) at the surface and likewise for subsequent rounds of reflectances. The total returning radiation is then

(4kT4βL)(1 + αLβL+(αLβL)²+…)
which evaluates to
(4kT4βL)/(1−αLβL)

Of this the surface absorbs a share (1-αL). Thus the total energy absorbed at the surface is

σ(1-αS)(1-βS)/(1−αSβS) + (4kT4βL(1-αL))/(1−αLβL)

For equilibrium this has to equal the energy radiated away from the surface; i.e., 4kT4. Thus energy balance at the surface then requires

4kT4 = σ(1-αS)(1-βS)/(1−αSβS) + (4kT4βL(1-αL))/(1−αLβL)
or, with a transposing of the last term
4kT4 − (4kT4βL(1-αL))/(1−αLβL) = σ(1-αS)(1-βS)/(1−αSβS)
which reduces to
4kT4[1 − βL(1-αL)/(1−αLβL)] = σ(1-αS)(1-βS)/(1−αSβS)
and, upon multiplying by
(1−αLβL) reduces
still further to
4kT4[(1−αLβL) − βL(1-αL)]
= σ(1-αS)(1-βS)[(1−αLβL)/(1−αSβS)]
and yet further to
4kT4(1 − βS) = σ(1-αL)(1-βL)[(1−αLβL)/(1−αSβS)]
and hence finally to
4kT4 = σ(1-αL)[(1-βL)/(1-βS)][(1−αLβL)/(1−αSβS)]

Let T0 be the surface temperature with no clouds or greenhouse effect and TC the surface temperature with clouds but no greenhouse effect. Then

4kT04 = σ(1−αS)

Then this with the previous formula for TC imply

(TC/T0)4 = [(1-αL)/(1-αS)][(1-βL)/(1-βS)][(1−αLβL)/(1−αSβS)]

The albedo of Mars is about 0.25. Without clouds, vegetation and water that would be also the albedo for short wavelength radiation for Earth and also for Venus. The albedo of Venus is about 0.75 and this is an appropriate value for the cloud albedo for short wavelength radiation. Thus αS=0.25 and βS=0.75.

Note that if αLS and βLS then (TC/T0)=1.

However, if αL=0.1 and βL=0.5 then

(TC/T0)4 = (0.9/0.75)(0.5/0.25)[(1-0.05)/(1-0.1875)
= (1.2)(2.0)(1.17) = 2.8
and thus
(TC/T0) = 1.294

Thus for these values of the albedos the cloud cover would raise the absolute temperature by almost 30 percent. In general, if αLS and βLS then TC>T0.

(To be continued.)

Conclusion

If α and β denote the albedos of the planet surface and cloud cover, respectively, and subscripts of S and L denote short and long wavelength, respectively and TC is the absolute temperature of the planet surface with a cloud cover but no greenhouse effect and T0 the planet surface temperature with no cloud cover and no greenhouse effect then

(TC/T0)4 = [(1-αL)/(1-αS)][(1-βL)/(1-βS)][(1−αLβL)/(1−αSβS)]


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