Thursday, July 28, 2011

Climate and the Moon

In a prior post I criticize an article on climate which states that current climate models ignore the effect of conduction (i.e. direct contact) and convection, and focus exclusively on the greenhouse gas radiative effect. A comment to that post needs a full response. The comment in full is here:

“I almost bought your discussion until you provided the diagram from Kiehl and Trenberth which is ludicrous. The whole construct here is to create the illusion that the sun cannot heat the earth above minus 18 which is absolute nonsense based on assuming it is valid on geometric grounds to reduce the solar insolation by a factor of 4 then again by the albedo.

If this is valid how then does the surface temperature of the moon reach ~123 C - quoted by NASA.

And how do you explain daytime temperatures on Earth in excess of 50 C as has been recorded ?”

I will focus on the Moon part, just because it is the easiest, but it will be natural to see how to approach this for the Earth as well. First, I must point out the irony of the comment. In trying to defend the claim that the climate models ignore conduction and convection, and focus exclusively on radiation, the comment refers to a system (the Moon) where there is no atmosphere and thus no conduction (except within the ground itself) or convection! Second I have to wonder about how stupid the commenter thinks NASA is. Do they really think that scientists would consider models that are flagrantly in conflict with the most basic observation about the Moon’s surface (i.e. its temperature extremes)? Do they really think that scientists would come up with a calculation that Moon can’t achieve temperatures above, say, -18 C and then stare at 100 C temperature measurements and just leave the calculation as is for decades? Let’s consider how one would develop a model of the surface temperature of the Moon, and it will answer the objections raised in the comment, as well as outline how real science actually progresses.

The Average Blackbody Model

We start with a very simple model of a spherical body out in space receiving input from the Sun to the tune of 1400 W/m2. At the same time, the spherical body emits radiation with a power per square meter dependent on T4 (i.e. the blackbody law).

Notice that in this very simple model we are assuming several things:

1. that the conduction within the body is instantaneous, thus the temperature of the body is uniform, and the output energy is uniform.
2. the body is not rotating, so the radiation it is receiving is constant
3. there is no atmosphere, thus no conduction or convection outside of the solid body
4. the albedo is the average albedo of the Moon, or a=0.14. Thus this object absorbs 86% of the radiation coming in
I’m not saying this is a good model, but it is a simple one that helps one understand some of the concepts at hand. We will see shortly that it has a number of shortcomings, but for now we’ll see how far we can push it. This is a traditional procedure in science. You start with the simplest model you can write down, push out all the consequences you can until the model breaks, and then introduce the needed complexities to address those consequences (and no more!). Thus, you always have the simplest model that is consistent with as many of the observations that you can.

The total energy out of the body would be the blackbody term shown, multiplied by the total surface area of the sphere: the radiation is outward in all directions. The incoming radiation, however, strikes only one side. Further, it strikes at different angles. Applying calculus one finds that the effective area is simply the cross-sectional area of the sphere, or the area of a circle the same size as the sphere.

We can then write down the change in the temperature, which depends on the material (the mass and specific heat), given the net energy input to the body. I’ll call this dependency K...its exact value, although calculable, will not be important in the model except qualitatively. We then have

When the “energy in” is greater than “energy out”, the temperature increases. When “energy in” is less than “energy out” the temperature decreases. Once it reaches equilibrium, temperature remains constant. What constant? That would be when dT/dt=0, or...

If we look up the values for the actual Moon we get the following:

So our model is not too bad, for the average value, but it could probably be improved. So, now back to the comment which motivated this all:

“If this is valid how then does the surface temperature of the moon reach ~123 C - quoted by NASA.”

The bottom line here is that, if there is an observation that is in conflict with a model, one of the assumptions of the model is probably incorrect, or perhaps you’re comparing the wrong observations to the model. We assumed that the object has a uniform temperature but we know from the 3 data points above (the min, max and mean temperatures) that this is not true! Essentially our model didn’t even attempt to describe temperature variations on the surface, so it comes as no surprise that it is not consistent with them. Many times theorists will use a model, with known deficiencies, because they are interested in different questions: perhaps we are only interested in the average value, and what happens from that average value? In that case, it doesn’t make a lot of sense to include complexities that will be averaged out anyway when we want to answer our question.

To miss this point is to miss the entire process of comparing theory with experiment. It turns out, however, in this case we can make a few simple modifications to explore some of the temperature variation.

The Infinitely Slow Surface-Conduction Model

We use the same assumptions as before, with one modification (in bold):

1. that the conduction within the body takes an infinite amount of time (i.e. no surface conduction at all). Thus, each patch of surface acts as its own independent object
2. the body is not rotating, so the radiation it is receiving is constant
3. there is no atmosphere, thus no conduction or convection outside of the solid body
4. the albedo is the average albedo of the Moon, or a=0.14. Thus this object absorbs 86% of the radiation coming in
We consider two types of patches: one on the near side and one on the far side.

Near-side patch

Imagine a patch of surface 1 square meter, with the same albedo as the Moon (i.e. absorbing 86%), and a combined mass and specific heat summarized by a constant K’. The energy equation then becomes

at equilibrium we thus have

which again, is reasonably close to the real value. Notice that all we had to do is change the conduction assumption to get surface temperature variation. If you’re concerned that the maximum temperature predicted is lower than the observed, notice that I am using the average albedo of the Moon. There are parts of the Moon’s surface with a lower albedo, and will thus get hotter as a result.

Far-side patch

Now imagine a patch of surface 1 square meter, with the same albedo as the Moon (i.e. absorbing 86%), and a combined mass and specific heat summarized by a constant K’ but with no sunlight at all coming in. The energy equation then becomes

The only equilibrium value for this is T=0. If no energy is coming in, and we have energy going out, the object will keep cooling. So in this model we have the near side T=380 K and the far side T=0 K, at equilibrium.

Although the model is clearly wrong, it demonstrates one thing: you can easily get temperatures above and below the average-model calculation simply by having not all parts of the surface heated equally, and some non-zero time of energy “communication” (i.e. surface conduction or, if you have an atmosphere, conduction and convection with the atmosphere) between the parts. The extreme version calculated here simply demonstrates the effect and is not meant to be realistic.

Adding a Few More Complexities - Qualitative Discussion

We now modify the assumptions as follows

1. that the conduction within the body takes a finite, non-zero, amount of time
2. the body rotates once every 28 days, like the Moon
3. there is no atmosphere, thus no conduction or convection outside of the solid body
4. the albedo is the average albedo of the Moon, or a=0.14. Thus this object absorbs 86% of the radiation coming in
Although one could set up a simple calculation, or numerical model, to handle this case I am not going to go through the exercise. However, there are two effects that will happen when adding these two changes:

1. the maximum temperature predicted will be a bit lower than the no-conduction model. This is primarily because the moon rotates the near-side patch out of the the most direct sunlight relatively quickly. If the surface conduction is on the order of minutes, this will make little or no difference. If it is around hours to days then it will. In fact, one could use this difference to help determine the time constant (related to the constant K’) for the surface of the Moon
2. the minimum temperature predicted will be higher than the T=0 predicted from the no-conduction model. This is both because the moon rotates the far-side patch out of the dark, and that energy from the previously warmed surface will conduct to the far-side patch.
It is fairly straightforward to get a model that is nearly consistent with the observed temperature range, and is consistent with the thermal properties of the surface of the Moon. One can get even more careful by modifying assumption (4), and use the local albedo of the various patches. In addition, one would need to look at all patches on the near-side, taking into account the varying angle of inclination of the radiation. This will not modify the qualitative results.

Conclusions

This all started when I criticized someone’s commentary on climate models, where they claimed that the models exclude thermal conduction and convection, and thus the focus on greenhouse gasses was entirely inappropriate. A further comment claimed that these models put an explicit maximum temperature achievable when they calculate the surface temperature of an object from blackbody equations. The comment further criticized my use of the average model summary for the Earth for this reason.

Notice the procedure we employed to model the system, and address these concerns. We started with a very simple globally averaged model, and got an interesting temperature value similar to the data. We then added a few complexities, such a differential heating, and noticed how this gives a range of temperatures on the surface. We also noticed that the range was half right (half wrong?): the maximum was good, but the minimum was terrible. Adding rotation and non-zero conduction time gives some dynamics and can achieve reasonably close agreement. A more detailed implementation of the local albedo fixes the small errors, especially on the top end. By using this procedure, we can see exactly which parts of our model give which parts of the result. It also shows which parts of the model give the biggest effect, and which are there for small adjustments.

All that is needed to go beyond the average model, and achieve temperature well above the average, is to include differential heating of the surface and some non-zero time of energy “communication”. Once you heat different parts in different ways, and add rotations and time-delays of conductions, you get some interesting dynamics around the average, going both above and below the average. The average calculation is still useful, if you’re not interested in short-term dynamics. It is further useful as a pedagogical tool, because it is a lot simpler. Thus it is not “ludicrous” to use the diagram from Kiehl and Trenberth, as long as one is aware that this is a globally averaged model. If you attempt to infer things well beyond the point of the model, then do not criticize the model - criticize your comparison, and look for a more detailed model that addresses the questions that you’re interested in.

Wednesday, July 27, 2011

Knowledge and Belief

I was just directed to this announcement concerning an NSF survey on science literacy. The bottom line is that the NSF is deciding to change the wording of two questions in the survey. The original wording is “Human beings, as we know them today, developed from earlier species of animals,” and “The universe began with a huge explosion.” . The new wording is “According to evolutionary theory, human beings, as we know them today, developed from earlier species of animals” and “According to astronomers, the universe began with a huge explosion.” (emphasis mine). It is noted that there will be a transition period with the questions, with half of the surveys containing the new questions and half the old, to determine its effect.

The stated goal for this change, from the NSF, is to separate knowledge from belief. You might believe that humans are created in their present form, 6000 years ago, but the new questions try to ascertain whether you know that “evolutionary theory” says something different. Is this an important distinction? Is this what we really want to measure? Which is more important for a society? What is the difference between knowledge and belief?

It is quite clear that there will be at least one effect for this rewording: given that the US falls way behind other countries on science literacy, especially with these particular questions, the rewording will most likely increase these numbers with no other work done.

Definitions and Concepts

Beliefs are representations of the world. Specifically, they are representations that we hold to be correct for the real world...as opposed to hopes, which are also representations of the world by not ones that we hold to be necessarily correct. Along with beliefs we always have a confidence in the belief, specified as a probability (either explicitly or implicitly). Knowledge is simply that collection of beliefs that we hold with such high probability or, in other words, with such confidence that we do not significantly doubt them. The belief that the sun rises in the east each morning is considered knowledge for the reason that we hold it with an extremely high probability. This is not just as a result of an inductive reasoning process (i.e. it always has, in our experience, risen in the east each morning) but because it is part of a larger body of knowledge (i.e. astrophysics) for which it is just one consequence within a whole host of other well-established predictions.

Now, on to scientific literacy. The NSF defines scientific literacy as “knowing basic facts and concepts about science and having an understanding of how science works.” Why is it important? Again, the NSF: “It is valuable not only in keeping up with important science-related issues, but also in evaluating and assessing the validity of any type of information and participating meaningfully in the political process.” The question we must ask is, does the new wording measure scientific literacy better than the old wording? To do this, we need to outline the four possible types of people answering the two forms of the questions:

1. people who answer “yes” to the old and “yes” to the new
2. people who answer “no” to the old and “no” to the new
3. people who answer “no” to the old and “yes” to the new
4. people who answer “yes” to the old and “no” to the new
The wording change doesn’t change cases 1 and 2, adds case 3 to the “yes” category and it introduces the erroneous case 4. The cases can be summarized in another way, like

1. people who know both that, say, the universe began with a big explosion and that astronomers claim that this is true. This is indicative of scientific literacy.
2. people who don’t know, or do not believe, that the universe began with a big explosion and that don’t know that astronomers claim that this is true. This is indicative of scientific illiteracy.
3. people who don’t believe that the universe began with a big explosion but know that astronomers claim that this is true. (more on this below)
4. people who know that the universe began with a big explosion, but do not believe that astronomers claim that this is true. This might at first seem to be a totally unreasonable and marginal case, but I think it is more significant than perhaps is generally appreciated. These people might think that the new wording is a trick question (e.g. they might think that physicists, as opposed to astronomers, claim that it is true). I’ve had students answer questions in this way, so it is not quite as uncommon as one might think. These students overthink the problem: they know the fact, but are distracted by the extra complexity of the question, thinking that the test is trying to trick them.
Case 3: The Religious Believer

The only reason these particular questions were modified was because of the prevalence of religious belief. How do we know this? We don’t see a proposal to change “The Earth orbits around the Sun and takes a year to do it” to “According to astronomers, the Earth orbits around the Sun and takes a year to do it.” Why? Because no religion (now) has a stake in the answer to that question, and thus have no objection to the claim. Of course, if you go back to the days of Copernicus this was a different story and people were severely punished for too strongly making such a claim. The two questions that are proposed to be changed in this way are precisely the two concepts that crop up in every creationist tract, and are clearly the two major stumbling blocks for a literalist reading of the Bible or the Quran.

Aside from the motivation for the change, we can ask the question whether it is accomplishing something important anyway. Are these Case 3 people, who would answer “no” to the old question but “yes” to the new question, demonstrating scientific literacy? I don’t think so. What they’ve confirmed is that they know that some scientists claim that the universe began with an explosion, but they don’t believe it. This means that they don’t accept the data, or the methods, or both. If the question were about something on the fringes of science, then perhaps this is fine, but it isn’t the case with these two questions. Evolution theory, for example, is as well established as the Round Earth theory and the Germ theory of disease. To deny it is to deny all of the independent work in molecular biology, embryology, ecology, etc... which support it. Even though they may know that fact that biologists support Evolution theory, they have not demonstrated any scientific literacy in terms of “evaluating and assessing the validity of any type of information and participating meaningfully in the political process.” The same can be said of the Big Bang theory, to a slightly lesser degree (i.e. there isn’t quite the volume of completely independent fields of study supporting it, as there is for Evolution, but the data is nearly incontrovertible anyway). To deny either idea is akin to denying the Germ theory of disease.

Bottom line

Imagine someone answering “no” to the question “The world is round” but answers “yes” to “According to geographers, the world is round”. Would they be demonstrating scientific literacy? I don’t think so. Do we want to pander to the religious-motivated ignorance in this country, for the sake of increasing the appearance of scientific literacy? I don’t think so.

Tuesday, July 26, 2011

Science and Attitudes Toward Criticism

So this morning I got a strong criticism of my post, “The Not-so-Hidden Flaw in this Climate Argument”, which itself is a criticism of someone else’s criticism of a climate model (got all that?). I only had a very brief moment to look at the comment, but it put me in a good mood...a mood that I don’t think would be held by a similar-type criticism in a non-science arena. I think there is a very big difference between the way a scientist, through training, perceives and handles criticism which was exemplified with my mood this morning. Let me try to explain.

1. It is a very good day for a scientist to go in to work, and to demonstrate that one of his colleagues is wrong. It’s even better if the wrong idea/theory/model is one that is popular! For those scientists to adequately demonstrate that a popular idea is wrong, we have for them the Nobel Prize. Of course, it is very hard to demonstrate that a well-established idea is wrong because, by definition, a well-established idea in science is one where many many smart people have tried to show it is wrong and have failed. For those people who claim that scientists have a conspiracy to uphold popular scientific ideas (a criticism creationists level against the support of evolution), they completely miss the goals of every scientist.
2. It is also a good day when someone criticizes your idea. In the comment on my post, the criticism took the form of “if your idea is correct, how do you explain the following observation...”. Awesome! Why? First, someone bothered to read my post, and found it important/interesting enough to comment...that always makes me happy. Secondly, I’m now in a win-win situation. There are 3 possibilities:
• the criticism is flat out wrong, and I get a chance to both teach something, and to bolster my idea...I can be a bit more confident in my idea.
• the criticism is partly correct, and I get a chance to add a bit of nuance, or explore a part of my idea that I hadn’t fully considered
• the criticism is correct, and I have learned something about the world and have to modify my thinking (at the expense of scrapping my idea).
Each of these 3 possibilities is wonderful, and it put me in a good mood! In contrast, most people when criticized (think politics, sports, religion, etc...) get defensive. They don’t look forward to the possibility that they might be wrong, and may need to modify their thinking. I prefer the scientific perspective!

Now I need to go and address the criticism.