San José State University |
---|
applet-magic.com Thayer Watkins Silicon Valley & Tornado Alley USA |
---|
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.
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 α_{S} 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 kT^{4}, 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^{-2}K^{-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 kT^{4}(4πR²). For balance then
Thus the actual area of the planet is thus irrelevant and hereafter all computations will be per square meter of surface area.
Let β_{S} be the albedo for short wavelength radiation of a cloud cover. Then a share of β_{S} of the incoming radiation is reflected and a share of (1-β_{S}) passes through the clouds. For an atmosphere transparent to the incoming radiation the energy incident upon the surface is σ(1-β_{S}). A share (1-α_{S}) is absorbed and σ(1-β_{S})α is reflected away from the surface. This impinges upon the underside of the clouds and a share of β_{S} is reflected back down to the surface. The energy of this reflectance is σ(1-β_{S})α_{S}β_{S}. The next round of reflectances results in σ(1-β_{S})(α_{S}β_{S})² arriving at the surface. At the next round it is σ(1-β_{S})(α_{S}β_{S})³.
Altogether then the short wave energy arriving at the surface is
The surface absorbs a share (1-α_{S}) of this and thus the total energy absorbed by the surface from the short wavelength radiation is
On the other hand, a unit of surface radiates 4kT^{4} up to the underside of the clouds. If the albedos of the surface and clouds for long wavelength radiation are α_{L} and β_{L}, respectively, then the radiant of energy 4kT^{4}β_{L} comes back. A round of reflectances from the surface to the clouds and back again results in (4kT^{4}β_{L})(α_{L}β_{L}) at the surface and likewise for subsequent rounds of reflectances. The total returning radiation is then
Of this the surface absorbs a share (1-α_{L}). Thus the total energy absorbed at the surface is
For equilibrium this has to equal the energy radiated away from the surface; i.e., 4kT^{4}. Thus energy balance at the surface then requires
Let T_{0} be the surface temperature with no clouds or greenhouse effect and T_{C} the surface temperature with clouds but no greenhouse effect. Then
Then this with the previous formula for T_{C} imply
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 α_{S}=0.25 and β_{S}=0.75.
Note that if α_{L}=α_{S} and β_{L}=β_{S} then (T_{C}/T_{0})=1.
However, if α_{L}=0.1 and β_{L}=0.5 then
Thus for these values of the albedos the cloud cover would raise the absolute temperature by almost 30 percent. In general, if α_{L}<α_{S} and β_{L}<β_{S} then T_{C}>T_{0}.
(To be continued.)
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 T_{C} is the absolute temperature of the planet surface with a cloud cover but no greenhouse effect and T_{0} the planet surface temperature with no cloud cover and no greenhouse effect then
HOME PAGE OF Thayer Watkins |