I love AboveTopSecret. I’m not ashamed to admit it. Some people enjoy soap operas, some people love sex crazed vampires, some people love Monty Python movies. The difference with ATS is occasionally you get a gem of information. This morning was one of those. Running with the sudden debate over the accuracy of the math behind
global warming climate change, some guy who calls himself theredneck took it to a level way beyond my comprehension. And, from reading the East Anglia emails, beyond the guys who got Nobel Peace prizes for theirs. Since it is way beyond my comprehension, I’m just going to lift the entire post.
Thanks to buddhasystem, I have decided to finish some calculations I started some time back. I am posting them here. The following will be used:
- Due to character limitations, I will be avoiding the use of exponential expressions. I apologize for any difficulty this may cause; it causes me difficulty as well, but is an inherent weakness in the font systems used on the internet and tends to cause confusion itself when used.
- All values are given in metric units. The abbreviations used are:
- m = meter
- cm = centimeter (0.01m)
- km = kilometer (1000m)
- g = gram
- kg = kilogram (1000g)
- J = Joule
- kJ = kiloJoule (1000J)
- W = Watt
- s = second
- °K = degree Kelvin
Calculations, due to the size of the values involved in planetary mechanics, will be based on the km/kg/kJ units. Other units are used for conversion of physical values.
- The Kelvin temperature scale will be used. Remember that a degree Kelvim is equal to a degree Celsius; the two are interchangeable for purposes of temperature variance.
- All sources will, of course, be linked. This will, however, be done through the use of footnotes at the end and reference numbers, rather than by links embedded throughout the text, in order to keep the calculations themselves as uncluttered as possible.
It has been theorized that the use of anthropogenic (man-made) carbon dioxide is the reason for the recently observed warming trend from ca. 1960-1998. The present level of CO2 in the troposphere is stated by multiple sources as being on the order of 380 ppmv or 0.038% of the atmosphere. This represents an increase, based on the most liberal estimates I have uncovered for pre-industrial levels of 280 ppmv, of 100 ppmv or 0.01%. Since this base point is considered to be ‘safe and natural’, it would logically follow that any warming would have to be associated with the 0.01% increase and it alone.
All heat energy reaching the earth is from the sun, in the form of solar irradiance. Heatb reflected back into space is a result of this solar irradiance, and can therefore be considered the same in energy calculations. Solar irradiance can and has been quantified. The amount of energy reaching the planet is on the order of 1366 W/m². The planet presents a more or less circular profile to the sun, so the area of the earth normal to solar irradiance can be calculated as this circle. The earth is an average of 6371 km, with a troposhere layer surrounding it that averages 17km in height, which also must be included since it is the location of the atmospheric carbon dioxide. That means a circular area of
r = 6371 + 17 = 6388 km
A = π r² = π (6388)² = 128,197,539 km²
We can now calculate the amount of energy which is thus intercepted by the earth (including the troposphere):
1366 W/m² = 1,366,000,000 W/km²
1,366,000,000 W/km² · 128,197,539 km² = 175,117,838,274,000,000 W (equivalent to J/s)
175,117,838,274,000,000 J/s = 175,117,838,274,000 kJ/s
That result in in Joules (or kiloJoules) per second. Since most climate predictions are based on much longer time intervals, I will now calculate how much energy would be available during such a longer time interval such as the commonly used 100-yr. period:
100 yr = 36,525 days = 876,600 hr. = 52,596,000 minutes = 3,155,760,000 s
We can now multiply this time interval by the rate of energy influx to obtain the total energy that the planet will recieve from solar irradiation over the next 100 years:
175,117,838,274,000 kJ/s · 3,155,760,000 s/100yr =
Now we must calculate exactly how much of that energy will be affected by the increase in the amount of carbon dioxide in the troposphere. Remembering that the increase from pre-industrial levels is 0.01% of total atmospheric volume, we multiple this total energy by 0.0001:
552,629,869,311,558,240,000,000 kJ/100yr · 0.0001 =
55,262,986,931,155,824,000 kJ/100yr intercepted by anthropogenic CO2
Now let us turn to the question of how much energy is needed to increase global temperatures. Of course, the first and most obvious area to be heated is the troposphere itself. Air under average atmospheric conditions has a specific heat capacity of 1.012 J/g·°K and an average density of 1.2 kg/m³. The troposphere itself can be calculated by using the information presented earlier (average radius of earth = 6371 km and a troposhere extending 17 km above the surface). Thus the area of the troposphere can be determined by calculating the volume of a sphere of 6388 km radius and subtracting a sphere of 6371 km radius from it:
V(tot) = 4/3 π r³ = 4/3 π · 6388³ = 1,091,901,171 km³
V(earth) = 4/3 π r³ = 4/3 π · 6371³ = 1,083,206,917 km³
V = V(tot) – V(earth) = 1,091,901,171 km³ – 1,083,206,917 km³
= 8,694,154 km³
Now we can calculate how much energy it would require to raise the temperature of the troposphere by a single degree Kelvin:
1.012 J/g·°K = 1.012 kJ/kg·°K
1.012 kJ/kg·°K · 1.2 kg/m³ = 1.2144 kJ/m³·°K
1.2144 kJ/m³·°K = 1,214,400,000 kJ/km³·°K
Since our calculations are based on a single degree Kelvin temperature rise, we can write this as
1,214,400,000 kJ/km³ · 8,694,154 km³ = 10,558,180,617,600,000 kJ
But to be accurate, the troposphere is not the only thing warming up. It has been often claimed (correctly) that the oceans are a major heat sink. So let us now calculate the amount of energy required to raise the ocean temperature by a single degree Kelvin. The volume of water on the surface of the Earth is an estimation, but several estimations are available and all of them are close. Therefore, in the interests of conservatism, I am using the smaller of the estimated values: 1,347,000,000 km³. The specific heat capacity of water by volume is 4.186 J/cm³·°K at 25°C. Thus, in order to raise the temperature of the oceans by a single degree Kelvin:
4.186 J/cm³·°K = 4,186,000,000,000 kJ/km³·°K
4,186,000,000,000 kJ/km³·°K · 1,347,000,000 km³
= 5,638,542,000,000,000,000,000 kJ/°K
As before, since we are considering a single degree Kelvin temperature rise, this is equal to
We now add the values for the troposhpere and the oceans together to obtain the amount of energy required to raise the temperature of these two areas combned by a single degree Kelvin:
5,638,542,000,000,000,000,000 kJ + 10,558,180,617,600,000 kJ
= 5,638,532,558,180,617,600,000 kJ
Now, remember from earlier calculations the total amount of energy that is available from the solar irradiance that can intercept anthropogenic carbon dioxide:
So if we know the energy required to raise a single degree, and we know how much energy can be intercepted by the anthropogenic carbon dioxide, we can calculate how many degrees of temperature rise could possibly happen. Remember, please, that we are making the following assumptions in these calculations:
- We only include the energy required to raise the temperatures of the troposphere (where the carbon dioxide is) and the oceans (climatic heat sink). No energy calculations are included to this point for land masses or for upper atmospheric levels, each of which would, in reality, contribute in some way to the amount of energy required.
- We are assuming that 100% of the available solar irradiance is being absorbed by anthropogenic carbon dioxide. This includes shortwave solar irradiation which is actually reflected back into space without being absorbed, and it also includes radiation that is absorbed through other means such as photosynthesis.
- We are assuming 100% conversion of that intercepted energy by anthropogenic carbon dioxide into heat, and not calculating how much of that heat is dissipated back into space through emission.
All of the above are extremely conservative assumptions. Inclusion of them will only decrease the expected temperature increases due to anthropogenic carbon dioxide.
Now, the actual calculation we have been waiting for:
Energy(required) / Energy(available) = Ratio
5,638,552,558,180,617,600,000 kJ / 55,262,986,931,155,824,000 kJ = 102.03
It would require 102 times as much energy as is available to raise the temperature 1°K in 100 years.
In other words, if ALL of the solar irradiance that the anthropogenic CO2 could intercept were converted into heat, and if it took no energy to warm the land masses and the upper atmosphere, the temperature of the planet would only warm by about 0.01°K in 100 years.
Sleep well tonight. The sun will rise tomorrow.
Wow. Anyone care to rebut this math? It’s out of my league for sure.