How Earth Cools

As you may have seen at the end of my last post, I received an objection to the existence of a greenhouse effect on the grounds that it violated the thermodynamics of heat transfer, and if you read what it says it is essentially focused on heat conduction. The reason I am bothering with this post is that it is an opportunity to consider how theories and explanations should be formed. We start by noting that mathematics does not determine what happens; it calculates what happens provided the background premises are correct.

The objection mentioned convection as a complicating feature. Actually, the transfer of heat in the lower atmosphere is largely dependent on the evaporation and condensation of water, and wind transferring the heat from one place to another, and it is these, and ocean currents, that are the problems for the ice caps. Further, as I shall show, heat conduction cannot be relevant to the major cooling of the upper atmosphere. But first, let me show you how complicated heat conduction is. The correct equation for one-dimensional heat conduction is represented by a partial differential equation of the Laplace type, (which I would quote if I knew how to get such an equation into this limited htm formatting) and the simplest form only works as written when the medium is homogenous. Since the atmosphere thins out with height, this clearly needs modification, and for those who know anything about partial differential equations, they become a nightmare once the system becomes anything but absolutely simple. Such equations also apply to convection and evaporative transfer, once corrected for the nightmare of non-homogeneity and motion in three dimensions. Good luck with that!

This form of heat transfer is irrelevant to the so-called greenhouse effect. To show why, I start by considering what heat is, and that is random kinetic energy. The molecules are bouncing around, colliding with each other, and the collisions are elastic, which means energy is conserved, as is momentum. Most of the collisions are glancing, and that means from momentum conservation that we get a range of velocities distributed about an “average”. Heat is transferred because fast moving molecules collide with slower ones, and speed them up. The objection noted heat does not flow from cold to hot spontaneously. That is true because momentum is conserved in collisions. A molecule does not speed up when hit by a slower molecule. That is why that equation has heat going only in one way.

Now, suppose with this mechanism, we get to the top of the atmosphere. What happens then? No more heat can be transferred because there are no molecules to collide with in space. If heat pours in, and nothing goes out, eventually we become infinitely hot. Obviously that does not happen, and the reason becomes obvious when we ask how the heat gets in in the first place. The heat from the sun comes from the effects of solar radiation. Something like 1.36 kW/m^2 comes in on a surface in space at right angles to the line from the sun, but the average is much less on the surface of earth as the angle is at best normal only at noon, and if the sun is overhead. About a quarter of that is directly reflected to space, and that may increase if the cloud cover increases. The important point here is that light is not heat. When it is absorbed, it will direct an electronic transition, but that energy will eventually decay into heat. Initially, however, the material goes to an excited state, but its temperature remains constant, because the energy has not been randomised. Now we see that if energy comes in as radiation, it follows to get an equilibrium, equivalent energy must go out, and as radiation, not heat, because that is the only way it can get out in a vacuum.

The ground continuously sends radiation (mainly infrared) upwards and the intensity is proportional to the fourth power of the temperature. The average temperature is thus determined through radiant energy in equals radiant out. The radiance for a given material, which is described as a grey body radiator, is also dependent on its nature. The radiation occurs because any change of dipole moment leads to electromagnetic radiation, but the dipoles must change between quantised energy states. What that means is they come from motion that can be described in one way or another as a wave, and the waves change to longer wavelengths when they radiate. The reason the waves representing ground states switch to shorter wavelengths is that the heat energy from collisions can excite them, similar in a way to when you pluck a guitar string. Thus the body cools by heat exciting some vibratory states, which collapse by radiation leaving them. (This is similar to the guitar string losing energy by emitting sound, except that the guitar string emits continuous decaying sound; the quantised state lets it go all at once as one photon.)

Such changes are reversible; if the wave has collapsed to a longer wavelength when energy is radiated away, then if a photon of the same frequency is returned, that excites the state. That slows cooling because the next photon emitted from the ground did not need heat to excite it, and hence that same heat remains. The reason there is back radiation is that certain frequencies of infrared radiation leaving the ground get absorbed by molecules in the atmosphere when their molecular vibrational or rotational excited states have a different electric moment from the ground state. Carbon dioxide has two such vibrational states that absorb mildly, and one that does not. Water is a much stronger absorber, and methane has more states available to it. Agriculture offers N2O, which is bad because it is harder to remove than carbon dioxide, and the worst are chlorocarbons and fluorocarbons, because the vibrations have stronger dipole moment changes. Each of these different materials has vibrations at different frequencies, which make them even more problematical as radiation at more frequencies are slowed in their escape to space. The excited states decay and emit photons in random directions, hence only about half of that continues on it way to space, the rest returning to the ground. Of that that goes upwards, it will be absorbed by more molecules, and the same will happen, and of course some coming back from up there with be absorbed at a lower level and half of that will go back up. In detail, there is some rather difficult calculus, but the effect could be described as a field of oscillators.

So the take-away message is the physics are well understood, the effect of the greenhouse gases is it slows the cooling process, so the ground stays warmer than it would if they were not there. Now the good thing about a theory is that it should predict things. Here we can make a prediction. In winter, in the absence of wind, the night should be warmer if there is cloud cover, because water is a strong greenhouse material. Go outside one evening and see.

Science, the nature of theory, and global warming.

My summery slumbers have passed, but while having them, I had web discussions, including one on the nature of time. (More on that in a later post.) I also got entangled in a discussion on global warming, and got one comment that really annoyed me: I was accused of being logical. It was suggested that how you feel is more important. Well, how you feel cannot influence nature. Unfortunately, it seems to influence politicians, who end up deciding. So what I thought I would do is post on the nature of theory. I have written an ebook on what theory is and how to form theories, and while the name I gave it was not one that would attract a lot of readers (Aristotelian methodology in the physical sciences) it was no worse than “How to form a theory”. Before some readers turn off, I started that ebook with this thought: everyone has theories. For most, they are not that important, e.g. a theory on who trashed the letterbox. Nevertheless, the principles of how to go about it should be the same.

In the above ebook, I gave global warming as an example of where science has failed, not because we do not understand it, but rather the public has not really been presented with the issue properly. One comment about global warming is that scientists have not resolved the issue. That depends on what you mean by “resolved”. Thus one person said scientists are still working on relativity. Yes, they are, but that does not mean that what we have is wrong. The scientific process is to continually check with nature. So, what I want to do in some of my posts this year is try to give an impression of what science is.

The first thing it is not is mathematics. Mathematics are required, and part of the problem is that only too often scientists do not state clearly what they are saying, preferring to leave a raft of maths for the few who are closely in the field. This is definitely not helpful. Nor are TV shows that imply that theories are only made by stunning mathematics. That is simply not true.

The essence of science is a sequence of simple statements, which are the premises. For me, the correct methodology was invented by Aristotle, and the tragedy is, Aristotle made some howling mistakes by overlooking his own methodology. Aristotle’s methodology is to examine nature and from it, draw the premises, then apply logic to the statements to draw some conclusions, check with observation, and if the hypothesis still stands up, try to determine whether there are any other hypotheses that could have given equivalent predictions. Proof of a concept is only possible if one can say, “if and only if X, then Y”, in which case observing Y is the proof. Part of the problem lies in the “only”; part lies in seeing the wood for the trees. One of the first steps in analyzing a problem is to try to reduce it to its essentials by avoiding complicating features. This does not mean that complicating features should be ignored; rather it means we try to find a means of avoiding them until we can sort out the basics. If we do not get the basics right, there is no point in worrying about complicating factors.

To consider global warming, the first thing to do is put aside the kilotonnes of published data. Instead, in order to focus on the critical points, try modeling something simpler. Consider a room in your house in winter, and consider you have an electric bar heater. Suppose you set it to 1 Kw and turn it on. That will deliver 1 kilojoule of heat per second. Now, suppose doors are open or not open. Obviously, if they are open, the heat can move elsewhere through the house, so the temperature will be slower to rise. Nevertheless you know it will, because you know there is 1 kilojoule per second of heat being liberated.

The condition for long term constant temperature (equilibrium) is
(P in) – (P o) = 0
where (P in) is the power in and (P o) is the power out, both at equilibrium. This works for a room, or a planet. Why power? Because we are looking to see whether the temperature will remain constant or change, and to do that we need to see whether the system is changing, i.e. gaining or losing heat. To detect change, we usually consider differentials, and power is the differential of energy with respect to time. Because we are looking at differentials, we can say, if and only if the power flow into a system equals the power flow out is it at an energy equilibrium. We can use this to prove equilibrium, or otherwise, but we may have to be careful because certain other energy flows, such as radioactive decay, may be generated internally. So, what can we say about Earth? What Lyman et al. found was there is a net power input of 0.64 watts per square meter of ocean surface. That means the system cannot be at equilibrium.

We now need a statement that could account for this. Because the net warming effect is recent, the cause must be recent. The “greenhouse” hypothesis is that humanity has put additional infrared absorbers into the air, and these absorb a small fraction of the infrared radiation that would otherwise go to space, then re-emit the radiation in random directions. Accordingly, a certain fraction is returned to earth. The physics are very clear that this happens; the question is, is it sufficient to account for the 0.64 W? If so, power into the ground increases by (P b) and the power out decreases by (P b). This has the effect of adding 2 (P b) to the left hand side of our previous equation, so we must add the same to the right hand side, and the equation is now
(P in + P b) – (P o – P b) = 2 (P b)
The system is now not in equilibrium, and there is a net power input.
The next question is, is there any other cause possible for (P b)? One obvious one is that the sun could have changed output. It has done this before, for example, the “Little Ice Age” was caused by the sun’s output dropping with a huge decrease in sunspot activity. However, NASA has also been monitoring stellar output, and this cannot account for (P b). There are few other changes possible other than atmospheric composition for radiation over the ocean, so the answer is reasonably clear: the planet is warming and these gases are the only plausible cause. Note what we have done. We are concerned about a change, so we have selected a variable that measures change. We want to keep the possible “red herrings” to a minimum, so the measurements have been carried out over the ocean, where buildings, land development, deforestation, etc are irrelevant. By isolating the key variable and minimizing possible confusing data, we have a clear answer.

So, what do we do about it? Well, that requires a further set of theories, each one giving an effect to a proposed cause, and we have to choose. And that is why I believe we need the general population to have some idea as to how to evaluate theories, because soon we will have no choice. Do nothing, and we lose our coastal cities, coastal roads and coastal agricultural land up to maybe forty meters, and face a totally different climate. Putting your head in the sand and feeling differently will not cool the planet.

* Lyman, J. M. and 7 others, 2010. Nature 465:334-337.

Another Wellington storm

Yet another storm hit Wellington; this time winds were a mere maximum of 165 k/h (about 100 mph). Is this climate change? Whatever, it is interesting that climate change is now a major concern, which raises the question, what can we do about it? Suppose we answer, “Stop burning fossil fuels,” what would the effect be? Currently, the atmospheric concentration of carbon dioxide is about 400 parts per million (compared with about 280 ppm at the beginning of the Industrial Revolution). If we are concerned about the effects of such atmospheric carbon dioxide, then if we stop producing it right now, the 400 ppm remain. Now, as noted in the last post, the climate shows strong signs of what physicists call hysteresis. This is when the effect is something depends on how you got there, where the system has “memory” of previous times. In this aspect, the Greenland ice sheets are actually the last remnants of the last great Ice Age. As we heat the planet, all that happens first in some places is that ice melts, the extra heat being absorbed by the melting ice without any temperature increase. In other words, for a while what you see is not what you are going to get!

In my opinion, the major problem civilization is going to face is rising sea levels. If the Greenland Ice Sheet melts, then the sea will rise about 7 meters. Take a look at Google Earth and see what goes. Amongst other places, a significant fraction of Bangla Desh, and essentially all Pacific islands based on coral reefs (as opposed to the volcanic basalt peaks, but you cannot live on the side of them). So, how do you defend against that?

 One suggestion is to build sea walls. These would have to be around all the land, including alongside riverbanks, and they may have to last tens of thousands of years. And, of course, while you are making all the required concrete and moving rock, you are probably generating massive amounts of further carbon dioxide, which will lead to more of the Antarctic ice melting, thus cancelling any value from your efforts. You could build walls of up to fifty meters high, and that would certainly be adequate for as long as the walls last.

 You could try removing the carbon dioxide from the environment. At first sight this seems futile; there is just too much there. However, at least some can be removed without much effort if we regrow forests. You would have to start planting them, but once underway, they would happily consume carbon. Even more spectacular would be to grow marine algae. The kelps such as Macrocystis pyrifera are extremely fast growing, and you can harvest them by mowing them. I rather fancy collecting such kelp and using it to make either biofuel or other chemicals. The key is to ensure that the carbon is removed from the ocean.

 Currently, we produce about 10 billion tonne per annum of carbon dioxide. That means we have to remove 10 billion tonne per annum just to break even. It is unlikely we can do that, although what we can do, we should, so what other options are there? A massive deployment of nuclear power would slow the fossil fuel burning, but it would not remove any of the current 400 ppm, and who wants nuclear power?

 The simplest answer is for every tonne of water melted by the ocean currents, we deposit a tonne of snow into the ice sheets. That involves geoengineering, and the problem is, when you interfere like that with nature, the effects are probably not that readily calculated. Such proposals in the past have been met with opposition. The problem is, some countries are going to be adversely affected by the geoengineering, and these are the ones that, in the first place caused the problem. Of course if we do nothing, it is the Pacific Islanders and the Bangla Deshis who pay. Do we know what will happen if we intervene? No, we do not, but we know what will happen if we do not. Of course there is another problem: how do we decide, and who decides?