Global Warming: Do the Math

I have mainly limited my blogging to modern evolutionary theory and the clash between science and religion. The main reason is that these are the areas that I felt knowledgeable enough to make a contribution to the debate that is engendered around them. However, I have been reading about another interest of mine, global warming. It is my opinion that global warming is not only real but there isn’t even a scientifically valid doubt about it. It is bad news and I suppose I shouldn’t be so surprised by the resistance to accept it. But I am.

I am also dismayed at the resistance to accept it on the part of otherwise intelligent young people. Here are some comments from some young people whose opinions I normally respect:

I think the whole bit about Global warming is a load of shit

The funniest thing about the global warming crowd is that they rarely take into account that the earth is still coming out of the last ice age.

Global warming may be real, but it’s not as bad as they make it out to be, and there IS NO CLEAR EVIDENCE in support of anthropogenic global warming.

It’s about time for that, if you think about it. We’ve had what? about ten years or so of the “global warming” crap, and before that it was about a decade of “global cooling” crap? Sounds about right then.

These comments were taken from blogs on entitled: The Truth About Global Warming & NASA Scientists Predict Period of Global Cooling. This website is primarily for high school and college age students. I deliberately chose comments from people whose work I have read before and have in my opinion posted thoughtful posts. Unfortunately they seem to think that the science behind global warming is weak, poorly done, being done only for the grant money, or the overall picture contradicts itself. That is unfortunate. My generation deserves a lot of the blame for global warming with our profligate use of energy but it is there generation that will have to deal with the consequences, and closed-eyed denial and conspiracy theory mongering is not going to put them in good stead to handle it.

I would like to do my part to stem this denialism by presenting the case for global warming. In stepping up to the plate to do this I feel like I am on familiar grounds. The promoters of global warming denialism use many of the same techniques as do creationists. They misrepresent the real science. They claim that the scientific establishment actively censures their point of view. They point to a legitimate controversy in peripheral area (such as whether or not global warming will lead to a greater number and more destructive hurricanes) as casting doubt on the entire field. I have had a great deal of experience dealing with such techniques and I think I can contribute some here.

The first step is to present the basis of the theory in a manner that shows the theory follows from simple logic. In the case of global warming, I think the way to do that is to look at the energy that strikes the earth’s surface. We can use simple math to show that the observed rise in CO2 concentrations HAVE TO lead to a rise in the global temperature of the earth. After that is established I will present the evidence that the rise in CO2 concentrations is caused by the activity of man. Once this is done, then there should be no way around it … anthropogenic global warming is a fact that should be dealt with.

One complaint that I often hear is that we can’t predict next week’s weather so how can we predict the climate decades from now? This is a fallacy. Climate and weather is not the same thing. Climate is best understood as AVERAGE weather. And the average weather is very predictable.

As a kid I remember a George Carlin skit in which he played “the hippy-dippy weatherman, Al Sleet”. Al gives us his long-range weather forecast (quoted from memory):

“The year is going to start out cold. There will be a gradual warming trend through spring, becoming hot in the summer. That will be followed with a cooling trend in the fall before becoming cold again at the end of the year”.

However, strictly speaking this is not a weather forecast it is climate forecast and everyone in the temperate latitudes of the Northern Hemisphere knows it is correct. So much so, that the humor in the skit lies in its obviousness. Climate is very predictable.

Climate change is also predictable. The reason is that, unlike next week’s weather, we know the major factors that affect it. And when we talk about GLOBAL climate there is really only one factor that we need to look at … the average energy absorbed at the earth’s surface. Let’s look at that now[1].

Let’s begin in the stratosphere. If one were to measure the energy coming from the sun at the top of the earth’s atmosphere one would find that 1,370 watts per square meter hits the upper atmosphere ever second. So the strength of the sun’s output is 1,370 W/m2/sec. Of course not all the earth receives this energy input. Half the earth is in darkness. The further north one goes the more oblique are the rays of sunlight that hit that part of the atmosphere. Thus, if we average the solar input across the entire earth we find that it averages out to 342 W/m2/sec.

If the earth receives that much energy, then it MUST also emit that much energy or else it will instantaneously heat up until it does. Some of the energy is emitted before ever makes it down to the earth’s surface. 77 W/m2/sec are reflected back from clouds and aerosols[2] in the atmosphere. Another 30 W/m2/sec hits the earth’s surface but is reflected back into space by its albedo[3]. Thus, 107 of the sun’s 342 W/m2/sec gets reflected back into outer space before it can absorbed. That leaves 235 W/m2/sec that are absorbed by the earth. The breakdown of that is that 67 gets absorbed by the earth’s atmosphere, leaving 168 W/m2/sec to be absorbed by the earth’s surface.

That is the amount of heat that we get directly from the sun. If that were the only heat that the earth gets then the average temperature at the earth’s surface would be -19o C. Hopefully you can see that this is much colder than the earh actually is. The average temperature of the earth’s surface is about 14o C, about 33o higher. So where does all this extra heat come from? The answer is greenhouse gases.

What has been long known is that certain gases absorb long-wavelength radiation and then re-radiate it. The earth’s surface radiates about 390 W/m2/sec. Of that about 324 W/m2/sec gets re-radiated back to the surface by greenhouse gases.

I’m going to pause here for a moment to do a little housecleaning. If you have been following the story so far you will note that the earth’s surface gets 168 W/m2/sec directly from the sun and another 324 W/m2/sec re-radiated to it by greenhouse gases. That totals out to 492 W/m2/sec coming in. But I have said that the earth radiates 390 W/m2/sec going out. There is a difference of 102 W/m2/sec. Why doesn’t the earth instanteously heat up to the point that its thermal radiation matches that of the incoming radiation? The reason is that there are two other forms of heat loss that the earth’s surface has that I haven’t yet mentioned. For completeness, let me do so now.

Wind blows across the earth’s surface. As it does some of the earth’s heat is picked up. In climatology, these winds are called thermals and averaged across the surface of the earth they disappate about 24 W/m2/sec. And then finally some of the earth’s heat is used to evaporate water that will make clouds and lead to rain. This averages out to a disappation of about 78 W/m2/sec. Or in other words, totals out to our (formerly) missing 102 W/m2/sec. OK, housecleaning is done, let’s get back to the greenhouse gas story.

Note that most of the energy hitting the earth’s surface actually comes from greenhouse gases (324 W/m2/sec for greenhouse gases compared to 168 directly from the sun). So the greenhouse gas effect is very strong. It naturally follows then that a change in the concentration of greenhouse gases WILL lead to a change in the amount of energy at the surface of the earth and therefore a change in climate. There is no way around that conclusion.

But we can’t end the story there. So far we have said a lot about greenhouse gases, but we haven’s said much about CO2. CO2 is a greenhouse gas, but it is not the only one. In fact, from the standpoint of the amount of energy re-radiated back to the earth it is not even the most important one. That honor should be given to water vapor, H2O. About 36%[4] of the greenhouse gas effect on the earth’s surface is due to H2O.

This is not a new discovery. Climatologists have known about this for a long time. So then why do global warming climatologists stress CO2 so much and talk so little about water vapor? Is this some conspiracy to intentionally hide the evidence? Not at all. Global warming climatologists are interested in climate CHANGE. Human activity does very little to directly change the concentration of water vapor in the atmosphere. But they do a lot that directly changes the CO2 concentration in the atmosphere. And CO2 IS the second most important greenhouse gas directly causing about 9% of the greenhouse effect on the earth’s surface.

Climatologists refer to a change in CO2 concentration as a FORCING, and a change in water vapor concentration as a FEEDBACK. What do they mean by that? What they mean is that under normal conditions it is the temperature of the atmosphere that determines the amount of water vapor in the air, the higher the temperature the more water evaporates and the more water vapor will be in the air. That is a feedback.

But if the temperature goes up there is a neglegible effect on CO2 concentration. While if we raise the CO2 concentration (which we are doing; evidence to be presented later) then that will FORCE a rise in the temperature. If one has followed the discussion so far (and I complement you if you have) then one may see a sneaky little fact of global warming here … a rise in CO2 concentration will directly cause a rise in temperature, which will in turn cause an increase of water vapor in the air, and the increased water vapor will have a further effect on the greenhouse gas effect. Thus, only part of the overall effect of raising CO2 concentrations is caused by its direct greenhouse effect, it also indirectly contributes to global warming by increasing the water vapor as well.

Let’s pause here for a moment. Up to this point I have said very little that is controversial. Everything I have said is to the best of my knowledge not seriously contested by global warming advocates or by global warming denialists. Everybody agrees that a rise in atmospheric CO2 concentration will lead to a rise in global temperatures. Long established physics of radiation and simple math shows that. What I have tried to do (and do in detail) is to give you the background information to evaluate claims that will be controversial, if not with the scientists involved in the actual research at least with those who claim that global warming isn’t really what the scientists say it is.

Some global warming denialists claim that global warming isn’t really happening. For that to be true then we should see that CO2 concentrations are not really rising. So what does the evidence say?

The answer is CO2 concentrations are rising and rising rapidly. The above graph show that. Since the early 1950’s we have measured the CO2 concentrations directly in the air. This measurement is done daily at an observatory on top of Mauna Loa in Hawaii. This site is chosen because it is far from any industrial source of CO2 emissions which could contaminate the measurement. That is shown in the red part of the graph.

Prior to that if we want to know what the CO2 concentrations were we have to infer them from other measurements. The best way to do it is to find some “fossil” air, air from the time of interest that has not been contaminated by air from other tmes. It turns out we have just what the science needs. We have ice cores from both Antarctica and from Greenland. These ice cores have yearly banding patterns on them that we can use to precisely date the area we use. Inside the ice is air bubbles, and inside those air bubbles is the “fossil” air we need. The data is so robust (from both the direct measurements at Mauna Loa, and from ice core data) that no reasonable person can deny that CO2 concentrations have dramatically gone up. And if the CO2 concentration goes up then global temperatures MUST rise. All we have to do to see that is follow the math. More CO2 in the atmosphere the greater the greenhouse effect. The greater the greenhouse effect the greater the energy at the surface of the earth. The greater the energy at the surface of the earth the greater the temperature … no ifs, ands, or buts.

Some global warming denialists say that the rise in CO2 concentration is not manmade. What does the evidence say?

There are direct and indirect ways to answer this question and ALL of them point to humans being the cause. Let’s look at an indirect method first.

Human input into global warming comes mainly from burning fossil fuels, oil and coal. They are called FOSSIL fuels because they came from living organisms. Living organisms grow by using biochemical pathways that are controlled by enzymes. Enzymes are molecules that direct chemical reactions. They do this by moving appropriate molecules together. So we have molecules moving molecules. And to do this requires a source of chemical energy. With respect to energy usage, enzymes are not much different than most humans … they don’t want to use any more than is necessary. Thus, they would prefer to move lighter molecule than heavier ones. There are two stable isotopes of carbon 12C which has 6 protons and 6 neutrons, and 13C which has 6 protons and 7 neutrons. 13C is heavier than is 12C. Thus, enzymes do a better job of using 12C than they do of using 13C. Living things then have an enrichment of 12C over 13C. Therefore if the cause of the increase in CO2 concentrations in the atmosphere is due to humans using fossil fuels then we should be able to see that 12C enrichment in the atmospheric CO2 of today compared to the CO2 found in the “fossil” air. And we do.

The direct way to determine if the rise in CO2 concentration is due to human activity is look at the CO2 we emit into the air and follow it through the carbon cycle. Can it explain the rise?

We emit over 27 gigatons of CO2 per year[5]. The molecular weight of CO2 is 44. 12, or approximately 27% of that is due to carbon. Thus, a release of 27 gigatons of CO2 corresponds to a release of about 7.3 gigatons of carbon (GtC) into the atmosphere. That is our present carbon footprint and it is an all-time high.

The above diagram shows the carbon cycle. It shows how carbon gets distributed throughout the ecosystem. It uses data from 2006 when our carbon footprint was only 5.5 GtC. In short, the carbon we put into the air is easily enough to account for 100% of the observed rise in atmospheric CO2 concentration.

So the evidence strongly points to humans as the source for the increased CO2 concentration. Thus, if one actually believes what the science has to say, then there is little doubt that anthropogenic global warming is real. Humans are pumping CO2 into the air. It is causing a rise in atmospheric CO2 concentrations. We know from more than a hundred years of established physics that that rise in CO2 concentrations has to lead to global warming.

The above graph shows global temperatures over the past 1000 years. This is EXACTLY what we know MUST occur when there is a rise in atmospheric CO2 concentrations. This just confirms the math. For most reasonable people this enough. But global warming denialists would have you believe otherwise. I will end this post by very briefly looking at some of their arguments based on the above information.

The most common argument is that what we are seeing is a natural cycle. The present increase in temperature is due to an increased solar activity. But that ignores EVERYTHING above. There is no reason that increased activity would lead to a natural increase in atmospheric CO2 concentration. There is no reason that a natural rise in atmospheric CO2 concentration should appear to have any of the characteristics we see that strongly suggest it is due to human activity. The only way this theory flies is to ignore the science.

The second most common argument is that the specific models used by climatologists are not accurate. As it turns out they are very accurate, and as we learn quantitatively about factors like methane concentration, aerosol reflectivity, the indirect effects of CO2 rise (such as increasing H2O vapor), changes in the earth’s albedo by ice sheet melting, etc. the more accurate they become. But even if they weren’t everything presented in this post relies solely on well tested physics and observations that (to the best of my knowledge) are uncontested.

The final argument is that those damn scientists are doing it for the money. They are in it only for the grant money. Creationists make similar allegations about evolutionary scientists. Why do you think it is that these complaints ONLY come from people who do not like the results science comes up with. If one actually looks at the science one finds that there is controversies and these can become quite personal. The confrontations between the late Stephen Jay Gould and Richard Dawkins was legendary. Science is not a monolithic entity. One does not get ahead in science by agreeing with others. One gets grants by proposing research that is designed to find out what we do not know. That means that you have to point out the shortcomings of other people in the field. And scientists do just that. If one scientist has good reason to think that he can show the field has taken a wrong turn, then he has a surefire grant in the waiting. And if he turns out to be corrrect in that assessment then he has his career made.

Science isn’t in the good-old-boy business, it is in the evidence business. If your idea is wrong then it is unlikely that you are going to keep getting evidence supporting it … unless you fake your data. If you fake your data on a problem that plenty of people are working on, then someone who doesn’t care a whit about your pet ideas is going to find something that says you are wrong and then others are going be more than willing to “settle the controversy” and show that you are stupid and they are right. That is part of the self-correcting mechanism of science.

The final argument that I seem to hear is that Al Gore is an asshole, so global warming is a lie. This is a non-sequitur. Even if Al Gore were an asshole … and having listened to him, I don’t think he is … it makes no difference. Al Gore has taken the time to read the literature, which is something that the vast majority of his critics seem not to have done. But he has done none of the science. And global warming is based on science not on Al Gore.


(1) Much of the discussion that follows comes from the Intergovernmental Panel on Climate Change, Fourth Assessment Report, Frequently Asked Questions 1.1.

(2) Aerosols are small particles suspended in the earth’s atmosphere that are capable of reflecting sunlight. They are not constant. Humans emit aerosols in certain forms of pollutions, especially sulfur-containing solutions. These are generally washed out of the atmosphere within a few days by rain. The sulfur-containing aersols so washed out are major contributors to the acidity in acid rain. Volcanos can be a natural source of aerosol production. Particularly violent eruptions can shoot aerosols above the level of the highest clouds. In that case, gravity must bring the aerosols down to a level where rain can eliminate them from the atmosphere. Since they are often only slightly more dense than air and turbulent winds can stir up the stratosphere, this process can take weeks to years to complete. During that time period the earth will reflect more of the sun’s incoming light which will cause a temporary cooling event.

(3) Albedo is surface of the earth feature. The lighter the color of the earth’s surface the more of the earth’s sunlight it reflects back into space. This can potentially exacerbate the problem of global warming. As the earth heats up, more and more of the earth’s surface formerly covered by ice gets exposed. This darkens the earth’s surface causing it to absorb more of the sunlight causing it to heat up further.


Up until recently the United States has been the number 1 carbon emitter. China now has that honor. Unfortunately it isn’t because the United States has reduced its emissions, it is because China has significantly increased theirs.


5 responses to “Global Warming: Do the Math

  1. Amen, brother (or sister, as the case may be).

  2. Congratulations! Irrefutable and obvious. As everybody really knows. The one scientific parameter that is missing is that the human race has lost the ability to recognise and respond to an imperative. Believing things is a matter of fancy and in any case what we choose to believe doessn’t matter because we are part of the audience. Not the play.

  3. Here, it’s everything you need to know about global cooling

  4. Somehow i missed the point. Probably lost in translation 🙂 Anyway … nice blog to visit.

    cheers, Outscore

  5. You use the unit W/m^2/sec. A watt is already the unit J/sec. Does this mean that the sun radiates 1,370 J/m^2/sec/sec on to the stratosphere? That implies that every second the amount of energy per second the sun transfers goes up by 1370. I think you need to recheck your units you probably mean W/m^2, not W/m^2/sec. It is a bit petty but when you are talking about the maths behind global warming this kind of thing does matter.

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