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:

Degrees Centigrade = 303 - 273 = 30 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).

Summary

The incoming radiation from the sun is about 1370 watts per square meter as determined by the energy per second emitted by the sun reduced by the inverse square law at earth orbit. *We calculate the total absorbed energy intercepted by the Earth's disc (pi)r^2, its distribution over its surface area 4(pi)r^2 and take into account that about 30% of that is reflected back into space, so the effective radiation hitting the Earth's surface is about 70% of the incoming radiation reduced by four.**Radiative energy is equal to temperature to the fourth power by the Stefan-boltzmann constant.*However, the effective incoming radiation is also trapped by greenhouse gases and emitted down towards the surface of the earth (as well as emitted up towards space from this lower atmosphere called the troposphere), the most powerful greenhouse gas being CO2 (Carbon Dioxide) and most abundant and important is water vapour. *This doubles the radiation warming the surface of the planet. *The atmosphere is predominately Nitrogen gas (N2) and Oxygen gas (O2), about 95 percent. *These gases, however, are not greenhouse gases. *The greenhouse gas CO2, though only exists in trace amounts, and water vapour, bring the temperature of the Earth up from minus 18 degrees centigrade (18 below freezing) to an observed average of plus 15 degrees centigrade (15 degrees above freezing). Without these crucial greenhouse gases, the Earth would be frozen. *They have this enormous effect on warming the planet even with CO2 existing only at 400 parts per million. *It occurs naturally and makes life on Earth possible. *However, too much of it and the Earth can be too warm, and we are now seeing amounts beyond the natural levels through anthropogenic sources, that are making the Earth warmer than is favorable for the conditions best for life to be maximally sustainable. *We see this increase in CO2 beginning with the industrial era. *The sectors most responsible for the increase are power, industry, and transportation. *Looking at records of CO2 amounts we see that it was 315 parts per million in 1958 and rose to 390 parts per million in 2010. *It rose above 400 in 2013. Other greenhouse gases are methane (CH4) and Nitrous Oxide (N2O). Agricultural activities dominate emissions for nitrous oxide and methane. *A healthy earth is one that is in radiative equilibrium, that is, it loses as much radiation as it receives. Currently we are slightly out of radiative balance, the Earth absorbs about one watt per square meter more than it loses. *That means its temperature is not steady, but increasing.