The second global warming crisis?

This post starts with a little bit of science fiction. Suppose you wake up tomorrow and read in the newspaper that a scientific breakthrough led to the discovery of a cheap source of virtually unlimited energy, with zero greenhouse gas emissions, perhaps something exploring quantum vacuum energy. We can now celebrate victory and rest assured that us humans won’t be causing climate change anymore, right?

Wrong. Well, in fact we are impacting on the global environment at critic scales in many ways, but as far as our energy use is concerned there’s one problem that wouldn’t be resolved, or worse, would be created, if we had an unlimited energy source: the heat from energy use itself.

When you use your computer it heats up. When you use your car it heats up. Whenever energy is used in our daily activities, that energy is eventually transformed into heat, which is just a word for energy that is spread out randomly in uncontrollable degrees of freedom. That heat goes from your car/computer into the surrounding environment, and eventually spreads around, heating the planet by a tiny bit. Adding your energy use with my energy use and everyone else’s energy use those tiny bits add up to…? Good question: how much does the temperature of the Earth increase by global human energy use, and how much more energy can we use before we raise Earth’s temperature appreciably?

First we have to figure out how the Earth gets its temperature. The temperature of the Earth is approximately determined by how much energy it gets from the Sun, which depends on how hot the Sun is and how far the Earth is from it, much like the temperature of your marshmallow depends on how hot the fire is and how far the marshmallow is from it.

There is of course a comparatively small amount of heat coming from inside the planet. But it’s only a fraction of the solar energy flow. Current human energy use from marketable sources is around 16 TW from all sources combined (see graph and references at the end of this post), and approximately 91% comes from endogenous sources. This is of the same order of magnitude as the energy flow from the Earth’s hot core, which is approximately 44 TW. If for some reason the energy from the Earth’s center got released much faster, or if human energy consumption were raised much higher, however, the temperature of the Earth would of course increase.

Human consumption of solar energy won’t cause the planet to heat up because it is already part of the energy budget of the Earth. The sunlight would be absorbed by the soil and re-emitted as heat anyway (unless the solar panels absorb much more light than the natural surface they are placed on, but that effect should be small, since the average absorption on the Earth’s surface is already around 70%). The same goes for wind and biothermal, which are ultimately powered by the Sun. It is only “endogenous” energy use that will cause global warming, such as fossil (oil, coal, gas), nuclear (fission or fusion), and even geothermal (to the extent that it releases the heat that is trapped inside the Earth’s crust into the atmosphere faster than at natural rates).

Alright, so how much more power can we use before heating up the Earth appreciably? I used an admittedly simple model (see below for details), but which should give us a good enough estimate of the order of magnitude we are talking about. And I was surprised to find that it is much less than I first imagined. In fact, we cannot consume 100 times as much energy from endogenous sources as we do today without affecting the temperature of the Earth, regardless of the greenhouse effect. A 100-fold increase would raise global temperatures by approximately 1 oC. Increasing our energy use by a factor of 1000 would lead to a catastrophic 7 to 10 oC temperature rise.

What does that mean? First of all, clearly no-one should lose their sleep worrying about this effect before we sort out our greenhouse gas emissions. At the present rate, it would take hundreds or even thousands of years to increase our power use that much, and with present sources the greenhouse gases would melt our planet much before that. But it is interesting to realise that there is a hard cap on how much endogenous energy we can use regardless of the greenhouse effect. So we go back to the science fiction story of the start of the post: what if we did find a source of virtually unlimited energy that emits no greenhouse gases? Should we celebrate? I’m not sure. If there is one thing that seems to be certain, it is that if energy were 100 times cheaper, we would quickly find ways to use 100 times more energy than we do today.

The plateau of several of the most important non-renewable resources seems to be within decades or has already past. Some of those are just being burnt and irretrievably lost, but some materials like metals could in principle be recycled almost indefinitely with an unlimited energy source.  If we have a hard limit on global energy use, however, that also puts a limit on how much recycling we can do even in principle.

This would also seem to put a cap on some science-fictional scenarios about the future development of the human civilization on Earth. For example, the  fact that the cap is just two orders of magnitude away could mean that the so-called “technological singularity” can’t be much more than a steep growth followed by a plateau, all within hundreds of years from now (In the extremely optimistic science-fictional circumstance that we find an unlimited energy source with no greenhouse emissions that is. In fact, the estimated resource peaks place the overall plateau of global economic and technological development well within this century). Energy efficiency can sustain development for longer, but that would plausibly still be at most another order of magnitude in effective work per energy unit. We could veer into geoengineering, which is still a huge question mark as far as its side effects are concerned; and if we have learned something about human beings so far it is that we are not very good at predicting or mitigating the externalities associated with our activities.


Let’s consider a very simplified model of the Earth, just to get an idea of the order of magnitude we’re talking about. When talking about thermal radiation, physicists often refer to an ideal model called a black body. A black body isn’t necessarily black, mind you, only when it’s temperature is very low. It’s called a black body because it absorbs all the electromagnetic energy (sunlight, for example) that falls into it, but it eventually emits it back in the form of thermal radiation.

Now it turns out that by looking at how much energy (or more precisely, how much power—energy per unit time) a black body emits, we can figure out its temperature. According to the Stefan-Boltzmann law, the total power P of thermal radiation emitted per area A of a black body is proportional to the fourth power of its temperature T:

P/A=σT4.     (1)

Here σ=5.8×10-8 J/(s m2 K4) is the Stefan-Boltzmann constant. Don’t worry if you don’t understand the maths, just bear with me and you’ll get the message in the end. This equation tells us that the energy per unit time leaving an unit area on the surface of a black body at temperature T must be σT4 (what is an unit of time and of area depends of course on what units you use to write σ). But since the temperature of a black body is always positive, we can invert that equation to figure out its temperature if we know how much power it is radiating:

T=(P/σA)1/4. (2)

Now of course this is only for the power emitted as thermal radiation. If there is a high-powered laser beaming energy out of the body, for example, but little thermal radiation, we should not conclude that the temperature on the surface must be given by substituting the laser beam power on the Stefan-Boltzmann law. But if we ignore any coherent beams of energy and add the total thermal radiation leaving the body, the Stefan-Boltzmann law will give a pretty good measure of the temperature on the surface.

So let’s see what that tells us. The Sun is a good example of something close to a black body. It also emits radiation isotropically, that is, the same in all directions, or at least very approximately so. The sun is approximately a sphere, and the total surface area of a sphere is 4πR2, where R is the radius of the sphere. So the total power emitted by the sun must be

P=(σT4)(4πR2).     (3)

With an average temperature of T=5,778 K, and a radius R=6.7×108 m, this gives us a total thermal radiation power for the Sun of P=3.8×1026 W. Wikipedia tells me that this is the total power emitted by the Sun (its luminosity), so the Sun seems to fit very well the description of a black body.

The Earth isn’t a black body. First of all it doesn’t absorb all the radiation that falls on it. Some 30% of it is reflected on average (the measure of reflectivity is called the albedo). Also, it has an atmosphere, which creates the greenhouse effect. But suppose that instead of the Earth we had a black body of the same size and shape in its place, which absorbed the same amount of radiation from the Sun as the Earth does, and emitted it in form of thermal radiation.

The Earth is at an average distance of D=1.5×108 m from the Sun. The area of the Earth that is on the path of the sunlight is the area of its circular cross-section, πr2, where r=6.4×106 m is the average radius of the Earth. The sunlight power absorbed by the Earth is therefore 70% of the total sunlight power times the fraction of the spherical shell at distance D around the Sun that is covered by the Earth: πr2/4πD2. Plugging in those numbers we get a total absorbed power of 1.23×1017 W, and using Equation (2) (with A=4πr2) we arrive at a temperature for the Earth of around 255 K, which is approximately -18 oC. Now of course, the average surface temperature of the Earth is actually larger than that, it’s around 287 K = 14 oC, and the difference seems to be mostly because of the greenhouse effect.

We’re almost there. Now suppose that we add more heat to the Earth, except that now instead of it coming from the Sun, it’s coming from inside the Earth itself, from human energy use. The logic is still the same though: more power, more temperature, just as above. From equation (3) we can see that the extra power ΔP that would raise the Earth’s temperature (if it were a black body) from T0 to T0 + ΔT is

ΔP = σ [(T0 + ΔT)4 – T04] (4πr2).     (4)

Ok, so let’s plot that, for ΔT=1 K (note that a variation of 1 K is equal to a variation of 1 oC).

So for the temperature that we calculated above for the Earth (255 K), an extra 2×1015 W would be sufficient to raise the thermometers by 1 K. For a black body at the average temperature of the Earth’s surface, the number would be closer to 3×1015 W. The first number may be the most accurate, since that’s what we get before adding the greenhouse effect. In any case, I am interested in orders of magnitude here, so either way it’s somewhere around 1015 W.

The next question you may be interested in is: for a given initial temperature, if we keep increasing the power, how much will the temperature rise? We can also use Equation (4) to answer that question, and I’ll plot the result in another graph:

So if we added an extra power of 2×1016 , say, the temperature rise would be somewhere between a catastrophic 7 and 10 K according to this model.

Now to the important question: how much power do humans use on Earth, and how fast is it growing? The first question has a more or less easy answer. We currently consume somewhere around 1.6×1013 W from all sources combined (see below), and approximately 91% comes from endogenous sources. So the first conclusion that we get to is that we cannot consume 100 times as much energy from endogenous sources as we do today without affecting the temperature of the Earth, regardless of the greenhouse effect. This is somewhat surprising to me, as I imagined before doing the calculation that we would be many orders of magnitude below the threshold.

How long would it take us to get there? This website quotes the International Energy Agency as claiming that the world’s energy consumption will continue to increase at 2% per year. This would mean that the energy use would double every 35 years, and thus it would be 100 times higher in 230 years. According to the US Department of Energy, the energy use in the world has actually been like this between 1980-2007:

It’s not clear to me whether this is an exponential growth. Assuming that it is growing linearly in the long timescales, it would take thousands of years to reach the dreadful 2×1015 W mark.


2 thoughts on “The second global warming crisis?

  1. Off-topic
    I responded to your comment on Michael Nielsen blog but since I am in a tiff with Nielsen about political correctness my comment didn’t make it thru 🙂
    Here it is:
    @Eric Cavalcanti

    Cosma Shalizi is of the opinion that the the Singularity has already happened.

    As for may be colonizing other planets this has been pretty much debunked by Charlie Stross:
    (from the first comment on Michael Nielsen previous post “How hard is space travel, in principle?”)

    The “little problem” with this is that the proponents don’t really seem to grok how big (and empty…) is the Universe:

  2. Yes, I agree there’s a sense in which the Singularity is already happening. People tend to think of it as a particular point, but exponential growth looks the same from every point. I don’t think there’ll be one day when suddenly we will all lose grasp of the world as some radical singularitarians believe. Our time scales, our attention spans, are getting shorter, and I think it’s more likely that we will just keep adapting to it.

    And about colonising other planets, I don’t think we will have time to do that unless we solve our current crises. If we have millions of years of civilization ahead of us, I don’t see why it would be impossible, instead of just hard. What that article overlooks is that we may not need terrestrial planets, we may develop technology to terraform unfriendly planets to our needs. But that’s all speculation anyway; our civilization may not even have hundreds of years to spare, and that’s certainly not enough.

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