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Climate Science
07-11-2014 How Climate Works

Model

As climate science is a new science, there are many models for the climate and I learned my climate science at MIT in a free online edX course. One can generate a basic model for climate with nothing more than high school algebra using nothing more than the temperature of the sun, the distance of the earth from the sun, and the earths albedo, the percent of light it reflects back into space.

The luminosity of the sun is:

L_0=3.9E26 J/s

The separation between the earth and the sun is:

1.5E11 m

The solar luminosity at the earth is reduced by the inverse square law, so the solar constant is:

S_0=3.9E26/4(pi)(1.5E11)^2 = 1,370 watts/square meter

That is the effective energy hitting the earth per second per square meter. This radiation is equal to the temperature, T_e, to the fourth power by the steffan-bolzmann constant, sigma. T_e can be called the effective temperature, the temperature entering the earth.

S_0 intercepts the earth disc, (pi)r^2, and distributes itself over the entire earth surface, 4(pi)r^2, while 30% is reflected back into space due to the earths albedo, a, which is equal to 0.3, so

(sigma)(T_e)^4 = (S_0/4)(1-a)

from (1-a)(S_0)(pi)(r^2)/4(pi)(r^2)

But, just as the same amount of radiation that enters the system, leaves it, to have radiative equilibrium, the atmosphere radiates back to the surface so that the radiation from the atmosphere, (sigma)(T_a)^4 plus the radiation entering the earth, (sigma)(T_e)^4 is the radiation at the surface of the earth, (sigma)(T_s)^4. However,

(sigma)(T_a)^4=(sigma)(T_e)^4

and we have:

(sigma)(T_s)^4=(sigma)(T_a)^4 + (sigma)(T_e)^4 = 2(sigma)(T_e)^4

T_s=(2^(1/4))(T_e)

(sigma)(T_e)^4=(S_0/4)(1-a)
sigma = 5.67E-8
S_0=1,370

(1,370/4)(1-0.3)=(1,370/4)(0.7)

S_0=239.75

(sigma)(T_e)^4=239.75

(T_e)^4 = (238.75)/(5.67E-8) = 4.228E9

T_e=255 degrees kelvin

So, for the temperature at the surface of the Earth:

(sigma)(T_s) = 2(sigma)(T_e)^4

T_s=(2^(1/4))(T_e)

or

T_s = 1.189(255) = 303 degrees Kelvin

Lets convert that to degrees centigrade:

And, lets convert that to Fahrenheit:

Degrees Fahrenheiht = 30(9/5)+32=86 Degrees Fahrenheit

In reality this is warmer than the average annual temperature at the surface of the earth, but, in this model, we only considered radiative heat transfer and not convective heat transfer. In other words, there is cooling due to vaporization of water (the formation of clouds) and due to the condensation of water vapor into rain droplets (precipitation or the formation of rain).

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