Energy, the Environment, and What Mathematicians Can Do (Part 2)

A couple of days ago I begged for help with a math colloquium talk I’m giving this Wednesday at Hong Kong University.

The response was immediate and wonderfully useful. Thanks, everyone! If my actual audience is as knowledgeable and critical as you folks, I’ll be shocked and delighted.

But I only showed you the first part of the talk… because I hadn’t written the second part yet! And the second part is the hard part: it’s about “what mathematicians can do”.

I include just one example of what you’re probably dying to see: a mathematician proving theorems that are relevant to environmental and energy problems. And you’ll notice that this guy is not doing work that will directly help solve these problems.

That’s sort of on purpose: I think we mathematicians sit sort of near the edge of the big conversation about these problems. We do important things, now and then, but their importance tends to be indirect. And I think that’s okay.

But it’s also a bit unsatisfying. What’s your most impressive example of a mathematically exciting result that also directly impacts environmental and energy issues?

I have a bunch of my own examples, but I’d like to hear yours. I want to start creating a list.

(By the way: research is just part of the story! One of the easier ways mathematicians can help save the planet is to teach well. And I do discuss that.)

What’s your most impressive example of a mathematically exciting result that also directly impacts environmental and energy issues?

There are a lot of historical examples, of course, and I doubt that there are examples where some mathematical discovery had some impact directly, without the usual time span it takes to figure out what to do with it…

What about wavelets?

Wavelets were developed to analyze time series in geosciences, as a means to separate effects with long range correlations that act on widely different length and time scales. It took some time until mathematicians discovered that they were doing quite similar things in harmonic analysis. The topic evolved around a close interdisciplinary collaboration, highlights include

* on the math side: the discovery of the Deaubechie wavelets and other functions that introduced new “special” functions into mathematics on par with the Bessel functions etc.,

* on the application side new algorithms for the analysis of wavelets and the compression of data, inluding the file formats JPEG for images and MP3 for sound, for example.

No, but of course there are examples where weather and climate data time series both from observations and from climate models have been analyzed with wavelets.

Here is an educated guess: spectral methods seem to be popular for the global solution of (an appropriate approximate version of) the Navier-Stokes equations for the atmosphere. A somewhat older reference for this kind of work is

* T.N. Krishnamurti, H.S. Bedi, V. Hardiker, Leela Watson-Ramaswamy: “An Introduction to Global Spectral Modeling” (Springer, 2nd edition 2010)

Spectral methods are about approximation in certain function spaces, for global approximations one uses of course sperical harmonics. Now there are various reasons that come to my mind why wavelets aren’t used here (unless they are, of course):

1. people think it is to hard to implement a method using wavelets,

2. people did not think about this before, because wavelets are considerably younger than the code snippets used to simulate flow dynamics,

3. people are satisfied with the existing solution and concentrate on other problems.

Actually I think it is a combination of all three. It’s the points 1 and 2 where mathematicians with some affinity to programming could help. IMHO it would be worthwhile to think about climate models with adaptive methods, that increase and decrease the local model resolution within the bounds of what is computationally feasable, depending on a metric that measures if something interesting is happening somewhere (formation of a storm, say). Of course I would be very glad if someone who knows more about this stuff and reads this would care to comment.

– Marta de la Llave Plata and Stewart Cant: “On the Application of Wavelets to LES Sub-grid Modelling” (in Pierre Sagaut, Bernard J. Geurts, Johan Meyers (editors): Quality and Reliability of Large-Eddy Simulations, Springer 2008)

Naively, there does not seem to be a reason why this cannot not be useful for climate modeling.

Left turns being the American and European version of right turns for Britain and Australia, and presumably other countries…:) I was a little confused for a moment as to why one’s engine would idle in turning left, but then it struck me.

21st-century mathematics may be driven by our need to understand the biosphere and our role in it.

I would say “should”: If mankind still has a grain of scientific (as opposed to technological) pride like in gone centuries, we should at least try hard to understand what we’ve ruined. Anyhow it is beautiful stuff, and beauty is worth study on it’s own right. But we can also expect that a deeper understanding will have practical benefits. (Just like General Relativity etc. has meanwhile very practical applications in my mobile navigation system.) Moreover I expect deeper philosophical enlightenment: The old world of math and physics was a world made of dead matter. Now we should care about living matter. E.g. to make Lovelock’s Gaia theory rigorous (perhaps deduce it from thermodynamics) would be a great milestone in the history of natural philosophy.

The old world of math and physics was a world made of dead matter. Now we should care about living matter.

Good line. Also, the old world was concentrated (energy) and the new world is diffuse (entropy). Most likely that many near-term forms of energy are based on dispersed phenomena such as wind, solar, bio, etc.

In fact I may change it to “will”. When you say “you should study X”, it makes academics want to fight back. But if you say “everyone will be studying X”, it’s more likely to make them want to go along. Especially if it’s true.

Fori, Well, that’s assuming that the biosphere is generally like an equation… Net-energy systems might be energetic processes of complex organizational accumulation though, often having parts that are actively learning as they go, rather than equations.

If so, then to find equations to help work with them you’d need to study something *other than* how to think of them as equations… Rosen and others have considered it simply as having two subjects. One is the natural world that behaves by itself that we study. Then the other is the world of our theories we tinker with, looking for what will help us interact with the one.

Phil, I shouldn’t go out on this limb in the presence of real math masters… Still: Methinks math is more than about equations. Perhaps we can model the interknit learning phenomena with n-categorical meta math, plug that into an infinite dimensional information geometric space (made of measures representing the spatiotemporal distribution of living matter), etc., and get some qualitative information of how things behave. Or get a general recipe for model numerics (perhaps like the Runge-Kutta/renormalization connection, cf. Butcher group for a taste of such math).

The perspective has been changing from “equations that have to be solved” to “maps of objects whose properties we have to study” in mathematics, you can see this if you follow the development of functional analysis from around 1900 (it is about solving differential and integral equations) to today. Today we don’t study solutions of Fredholm integral equations, but the properties of Fredholm operators instead, for example.

Flori, Well, yes that describes a strategy, and could be used with an object language for components like Tim suggests. BUT even if that were feasible and not just poetic, aren’t you still stuck in the same situation of having to learn how to understand complex systems from observing their behavior, and not from having programmed it?

Isn’t that the step everyone seems to be avoiding, learning how to understand complex systems from watching how they work rather than from knowing the rules they follow?

Phil, I try to follow what you’re saying, but it just isn’t clear. Your website is also muddled. I suppose the opinion you are expressing is just “a theory of how water flows is futile because we can never understand the precise structure which dictates the lifecycle of water particles”. (And at bottom, isn’t that precisely what you are suggesting?). Before we give any part of ourselves to a field, we first ask whether it might be interesting, and here our ignorance and prejudice is valued, here we decide whether we shall proceed or move on to another. I mean, we love films, say, and by what other process do we ever come to watch a film? I certainly have my own opinion–based on nothing but ignorance and taste–that meteorology (here i take the naive view that the business of meteorology is prediction) is bounded. That is, I’m amused by forecasters who credit the limitation of weather predictions to inadequacies in our models.

But on all this i’m an amateur and mildly embarrassed to fill the blog with such low quality ramblings. I would hope my comments are deleted–in the name of waste management.

Basically, I don’t understand what you’re saying. I’m under the impression that you think mathematicians believe themselves to possess some view of ”truth” in that ridiculous sense of old french philosophers.

I’m amused by forecasters who credit the limitation of weather predictions to inadequacies in our models.

Weather models model the atmosphere only, they basically monitor what get’s blown our way. But that still means that the horizontal grid resolution is around 5 km, which is not fine enough to resolve many cloud formation processes, for example. From the viewpoint of a single human on the surface, to have one grid point every kilometer isn’t impressive, right? But these are the limits due to limited computational resources. When you set up a global model, a quick back on the envelope calculation can tell you for example that combining a 50 km horizontal grid spacing with 25 vertical grid cells leaves you with 5 kB of memory per grid cell if you have 1 GB memory for grid data overall.

Simply put: There are a lot of relevant physical processes that cannot be resolved in weather and climate models, but have to be put in by hand via heuristic corrections, due to the coarse grid that is enforced by limited computer resources.

Phil, I’m not sure I understand your objection: You seem to say we can’t in principle understand much (if anything) of the workings of nontrivial systems – because that would require us to predict their inventions (reorganizing micro details, inventing new macro rules).

Indeed methinks that would require us to be all-knowing Laplace demons. It would be like wanting to predict detailed evolution of life forms.

(Who is “Rosen and others”? Quote/link?)

But I think we can understand qualitatively and partially how complex systems work. Perhaps the success of physics is an example for that: Nature is full of emergent properties where the exact micro details don’t matter and some “higher principle” is responsible, like maximum entropy production. Some emergent properties can be modelled mathematically – and that’s probably the reason for the unreasonable effectiveness of mathematical equations. That we know some fundamental constants with great precision is perhaps due to them describing emergent properties. Of course, observing and learning about complex system behaviour is an essential part of natural philosophy. There is perhaps not much difference between “dead matter” and living matter.

Here’s another example of “qualitative math”: Estimating the topology of spaces from their curvature (e.g. Betti numbers). Earth is finite because its curvature at any point is positive.

Mr. van Beek, suppose you had unlimited computational power ie. you could instantaneously measure every square meter of this planet. That is, suppose you had all the information you could possibly ever want. Would that not paralyze you?
Aren’t our models essentially based on the fact that we have limited and coarse input data?

Permit me to be vulgar here: what is the category within which meteorologists work? To rephrase, how would one be a weatherman for 4-manifolds?

Flori, Justin & Tim, sorry I didn’t notice your questions for a couple days. I’m not in the least “criticizing” models for being naturally incomplete, but suggesting we use that property to help expose and study the ways nature does more than we can model, studying how nature completes natural system behaviors by comparing model and observed behavior. I’m suggesting a forensic science approach, comparing standard models we understand well to help reveal coherent behaviors of natural systems that diverge in the model we understand.

No doubt that seems a muddleheaded way to construct a model, as it’s something of the opposite, not how to construct a model but how to deconstruct our observations of nature. Of course, the main “coherent departure” of nature from what we can model is the emergence of new systems.

There might be other clues to use, but my theorem of emergence and continuity proves that to satisfy energy conservation new energy using processes need to exhibit a period of “inflation”. It’s a conclusion that is quite broad, that the big bang’s period of inflation, found necessary to explain the astrophysical data, would be a special case of.

Now do you understand why I’d scan data looking for emerging processes of inflationary development?

You may call me Tim here, I don’t insinst on a formal introduction :-)

…suppose you had unlimited computational power ie. you could instantaneously measure every square meter of this planet. That is, suppose you had all the information you could possibly ever want. Would that not paralyze you?

Computational power and better input data are of course two separate topics, to improve weather models we need to improve both. The question how far we could go in principle is of course an interesting one, but when we take a look at weather models today from a practical viewpoint, it should be clear that there are a lot of obvious flaws and problems that can be solved/improved simply by more computational power, and more data. There are projects underway in both directions. As a weather modeller I’d like to incorporate lakes and cloud forming processes due to lakes into my model, for example. With a grid spacing of 5 km that is hardly possible.

I’d really like to know long a global weather forecast could be done that is of use (being deliberatly imprecise with this question). Today the longest model runs encompass ca. 16 days, I think.

Permit me to be vulgar here: what is the category within which meteorologists work? To rephrase, how would one be a weatherman for 4-manifolds?

I’m sorry, I don’t understand what “category” means in this context, nor what is vulgar about the question. To a first approximation global weather modelling is computational fluid dynamics on a rotating sphere, is that your point? I suppose that this becomes a lot more complicated in 4D…

For a decade and more my wife complained about word processing people, and I kept telling her: you’ll never get documents created as you want in a timely fashion unless you do it yourself. Programming is a bit like that too: and spreadsheet programming shows that people want to program if the environment is accessible enough. It is nice when sophisticated mathematics makes a difference. It is nice when real mathematicians can cooperate with people in other fields and use semi-sophisticated mathematics to help them. Yet I feel that the real need is for everybody to understand unsophisticated mathematics enough so that when they have a problem amenable to mathematical treatment they at least recognise that and look for help. The Internet seems a wonderful tool for educating people about mathematics, but I feel there won’t be much progress until the educational authorities stop regarding mathematics as an optional skill. At any rate one of the issues seems to be that mathematicians love to generalize, but the implications of the general theorems don’t percolate down. I remember an memoir by a famous mathematician (perhaps Arnold) complaining about papers published on how to solve particular types of PDE, when they were just special cases of a “well known” general theorem. Unless mathematicians are also involved in the real world, or at least the less unreal world, then they won’t know how their understanding can make the difference it should.

If plants keep their leaf pores smaller, that means that particular plant types can grow in drier conditions. At the edge (if rainfall stays the same) that means more tree covered areas which is a negative feedback on CO2. In other places I presume it means that you eventually get different trees (basically because trees which are adapted to relatively dry conditions lose out to trees that don’t have those adaptations where those adaptations aren’t necessary). So it’s tricky. And isn’t this typical of so much relevant stuff. You see a summary of a result and you wonder: “did they allow for this or that?”. It would be interesting to develop an oracle (like Watson?) which could absorb a lot of information and answer questions like that.

Thanks for the links, these are all examples of time series analysis or synthesis, for which wavelets were developed in the first place. A wavelet match of 20th century data based on some model for a random process could be used as a null hypothesis to test climate models, for example. In this sense it would be possible to use wavelets to synthesize time series that predict weather or climate data.

But when John asked

How about applications of wavelets to weather prediction or climate prediction?

I understood the question in a different sense, however, namely if wavelets are used as a tool in climate or weather models. Wavelets were invented to replace trigonometric functions, because people needed base functions that are localized both in time and frequency. Therefore I was speculating if wavelets could be used in global climate models to replace spherical harmonics in a spectral approximation, because then it would be possible to adjust the resolution of the approximation locally.

This is much like hp-FEM, or, to be more precise, hp-FEMs are an example of this method with a very simple wavelet basis (or, rather, a frame) consisting of localized polynomials.

Spectral methods are very popular because they allow the explicit control of the wave modes that are resolved in a model, which is precisely what you want from a climate model: Retain only the large scale effects. It would be interesting to know if you get different results when you enable the model to change its resolution locally adaptiveley.

There is one project mentioned on the page of Dr Irene M. Moroz which is close to what we were discussing:

We are interested in developing fast numerical algorithms for the dynamic wavelet Galerkin modelling of the two- and (ultimately) the three-dimensional incompressible Navier-Stokes equations for arbitrary boundary conditions, using compactly supported wavelets. Initially we have focussed our efforts upon developing a biorthogonal wavelet Galerkin formulation of the Burgers equation, which is a necessary first stage before attempting to address the two- dimensional Navier-Stokes equations.

Yes, that’s what I head in mind. I recently “discovered” Burgers’ equation as a simple model to start with myself, and have written a little bit about it here on the Azimuth wiki. Writing a Galerkin = spectral approximation for this equation is already a formidable homework.

Therefore I was speculating if wavelets could be used in global climate models to replace spherical harmonics in a spectral approximation, because then it would be possible to adjust the resolution of the approximation locally.

Yeah, that’s sort of what I had in mind: using wavelets to describe fluid flow, as part of simulating the weather. The paper you mentioned about large eddy simulation sounds like it’s heading in that direction, though I haven’t read it.

The problem is that I don’t know for sure either, especially if LES in the field of computational fluid dynamics is the same as LES in meteorology. The paper is about CFD, not about meteorology.

I forgot to mention that the climate and weather modeling books that I have mention “LES” as the most commonly used approximation technique for weather and climate models, but I’m not sure if they have the same understanding of what LES is as the CFD people who concentrate on engineering applications of fluid flow.

What’s your most impressive example of a mathematically exciting result that also directly impacts environmental and energy issues?

My example is not of a result but of an area of mathematics, namely random trees and branching processes. There are lots of examples in biology. The tree of life is important for understanding and evaluating biodiversity. Your body contains about 10^{}^{14} cells, each one produced by binary fission from a parent, back to a single zygote: that is quite a big tree. It contains sone interesting subtrees, such as the airways of your lungs, and models of those are useful for understanding the effects of air pollution for example. And the woody kind of tree, and their leaves can also be modelled by branching processes. From physics, there is the shower of particles (up to a billion) that can be produced when a cosmic ray hits the atmosphere – something which may have an important role in cloud formation.

A nice quote at the start, something Rota said before 1986:

“The lack of real contact between mathematics and biology is either a tragedy, a scandal, or a challenge, it is hard to decide which.” – Gian-Carlo Rota

Sturmfels’ work applying toric varieties to various areas in biology and chemistry is something I want to explain when I get back from Hong Kong and start talking about stochastic Petri nets.

As Tim suggested, there are a number of impressive, directly impacting developments in time series analysis. But wavelets, although nice, aren’t actually exciting – just special cases of spectral analysis.

I think the most exciting result in time series analysis was the discovery of universality in chaos (Feigenbaum et al.). I don’t know about the direct impacts, however.

I don’t know of any practical applications of chaos theory/ fractals, either. Wavelets, especially the discrete fast wavelet transform – went from the blackboard directly to the assembly line, that’s why I mentioned it.

Of course every physicist knows a lot of examples of mathematical discoveries that became important for applications – but usually much, much later.

But wavelets, although nice, aren’t actually exciting – just special cases of spectral analysis.

I guess we all get excited by different things, but back when wavelets first came out, I seem to recall a bunch of mathematicians being excited. If you google wavelets excitement you’ll get some interesting hits.

Here’s an intro for mathematicians, back from those days I guess:

It doesn’t capture a sense of ‘excitement’, but it’s nice and clear, at least for mathematicians who are already comfortable with Fourier analysis and want to learn about wavelets.

However, this article has nothing to do with energy, the environment, biology, etc.

Computers of course. The first televised weather program was shown here in Sweden first in september 1954 with Rossby and the BESK computer and then John von Neumann and R Charney did a televised forecast six months later using “his” ENIAC.

And the second must be topology, which has given us means to understand and predict nonlinear chaotic phenomena which we hopefully can be able to control as well. (Source: Eugenia Kalnay, Atmospheric Modeling, CUP 2003.)

Just a couple of quick observations about, sensors, signal processing, and climate change. Signal processing is at the heart of detecting “hidden” signals in noise, signals the are slowly varying in time, and anomalous behavior detection of phenomena using sensors. Wavelets have become increasingly important for different sensor applications including solving of PDE’s related to weather modeling and forecasting. Also, see
wavelet + radar + weather for detecting anomalous weather such as turbulence, tornadoes, etc. Wavelet have been widely used in ground penetrating radar, which have been used for detecting buried objects but can also be used detecting changes in soil characteristics for monitoring how climate zones change.

Another example is time-frequency signal processing methods which were pioneered by Leon Cohen of Hunter college in the paper: “Generalized phase-space distribution functions” – Journal of Mathematical Physics, 1966. This paper was considered foundational to signal processing work on time-frequency methods for signal processing that revolutionized examination of signals that change with time and recognizing them. While this is widely used in radar, sonar, and acoustical data analysis, it has found limited use in climate change per say. Climate data presented this way would probability make some aspects of change much more apparent as they have in other time varying signals generated by nature and detected by sensors.

This would be fun to do with as a student thesis project for those of you looking for ideas for student projects.

Ensemble Kalman Filters seem to be used a lot for data assimilation in weather forecasting, not sure if and how this translated to climate prediction, but here are a few links:

But I have seen more. Especially a lot of work in the systems biology area recently. Not to mention modeling and control of large scale electrical networks, variable speed wind turbines, and of course, Tokamak plasmas. They all have a potentially big impact on environmental issues.

If anyone is interested in one area in particular i can try to dig out more literature on Mathematical Modeling and Control in that specific area.

Ensemble Kalman filters (EnKF) have been used for parameter estimation problems in climate modeling. They are now gaining popularity for state estimation in decadal prediction problems, for the same reasons they are used in weather prediction: sensitivity of the system state to initial conditions.

Here are a few standard references to wavelets and signal processing.

Stephane Mallat, A Wavelet Tour of Signal Processing. Look at both the second and third editions which cover somewhat different material.

Jean-Luc Starck, Fionn Murtagh, and Jalal M. Fadili, Sparse Image and Signal Processing: Wavelets, Curvelets, Morphological Diversity, Cambridge 2010. Many up-to-date references.

Scientific Computation is emerging as absolutely central to the scientific method. Unfortunately, it is error-prone and currently immature: traditional scientific publication is incapable of finding and rooting out errors in scientific computation; this must be recognized as a crisis. Reproducible computational research, in which the full computational environment that produces a result is published along withe the article, is an important recent development and a necessary response to this crisis.

We have been practicing reproducible research for 15 years and integrated it with our scientific research, and with doctoral and postdoctoral education. In this article, we review our approach, how the approach has spread over time, and how science funding agencies could help spread the idea more rapidly.

That’s a very interesting article—thanks! I think Tim van Beek will be interested in it. I like this quote (a quote of someone in the article, not a quote of the article itself):

“an article about computational science in a scientific publication is not the scholarship itself, it is merely advertising of the scholarship. The actual scholarship is the complete software development environment and the complete set of instructions which generated the figures.”

I think the world-wide web is a prerequisite for being able to publish the actual scholarship. So before the web existed, there was an excuse for not doing this.

The title of Sturmfel’s paper is provocative and seriously disappointing. It seems very possible that mathematics is biologically impotent.

At bottom, I think our business as mathematicians is simple: we solve problems. We are attracted to question marks and contradictions. If biology is to attract mathematicians, then she must lure us with problems. And something beyond the usual, and tedious, and dull, and wholly predictable and tiring affairs of dynamical systems and population growths or of modelling weather. And if she has nothing to lure the young and intrepid, what then?

Biology is full of question marks and contradictions. It’s physics, not biology, that studies the “dynamical systems” you find tedious, dull, wholly predictable and tiring. Biology demands new ideas that we’re just beginning to have.

(By the way: if you think the subject of weather is dull, you either haven’t studied it very much, or your tastes are just fundamentally different than mine. The more I learn about it, the more interesting it gets.)

I misunderstood the main interest of this blog. The question of “mathematics and the environment” is not that of the relation between “mathematics and biology”. As a student not exactly interested in fluid dynamics (which i consider–without any real experience or justification–as the actual domain of meteorology and weather) i see here nothing compelling.

Sure, among experts, the questions are enthralling. But does it have anything to catch the eyes of those whom it has not already absorbed? I mean, something beyond wavelets. It seems to me that is the first problem of the whole field—it’s closed and hides itself from mathematicians. Does it find no stronger ally in topology than the statement that somewhere the wind is not blowing, or that somewhere barometric pressure and “something else” coincide?

I’m obviously ignorant, prejudiced, and skeptical. But I’m also a young energetic student who’s looking for something fascinating. And when I turn my way towards, for instance, this website and the comments, I find only the typical talk of applied mathematicians. I conjecture that a field is only significant if it is interesting geometrically. And so?

One may either dismiss everything i’m trying to say as the ramblings of an obnoxious kid, or one may try to catch the song of a rare bird flying around this field, and playing it in the ears of those passerby’s who are looking around, trying to find something to tap their foot to.

Note: if the interest of this blog was actually biological, I would begin by referring the readers to the article by Gromov and Carbone titled “Mathematical Slices of Molecular Biology”.

Sure, among experts, the questions are enthralling. But does it have anything to catch the eyes of those whom it has not already absorbed?

I’m no expert. I may however find it easier than you to become interested in things. I often wish I had days that were twice as long, and twice as much physical energy, to learn about all the fascinating things in this world. I have a bunch of books on my desk: Fundamentals of Chemical Kinetics, and The Theoretical Biologist’s Toolbox, and Wetlands Ecology: Principles and Conservation, and Chemical Reaction Networks: A Graph-Theoretic Approach. They’re all really cool, but unfortunately I don’t have time to read them all, because I want to keep explaining network theory and how stochastic Petri nets can be understood as a variant of quantum field theory in which probabilities replace amplitudes. Since stochastic Petri nets are also related to toric varieties, there should be some fun relation between quantum field theory, or at least this variant, and toric geometry. And I want to figure out what it is!

Note: if the interest of this blog was actually biological, I would begin by referring the readers to the article by Gromov and Carbone titled “Mathematical Slices of Molecular Biology”.

What I’m pointing out is that inventing your own models of natural systems to study, as physics relies on, has major drawbacks. It represents nature as designed according to your own modeling technique and value system, that you embed in constructing your model. So you’re not studying your subject in its natural form at all, and having no way to discuss what the significant difference between nature and model might be. You only get to discuss the significance of differences between model A and model B.

It results in your only studying imaginary systems, is the simple way to state the basic dilemma. It means losing all interest in how natural systems have fluidly changing organization and widely distributed animating parts, causing you to fail to look for or find the questions about such things that actually are answerable. Studying only imaginary systems diverts interest from the study of the natural processes themselves.

There are an enormous range of things that artificial systems can’t represent that are quite common behaviors and organizational states found in uncontrolled natural systems in open environments. That appears to be a central reason for why modeling has not been of much help for understanding them.

Granted, you need to look at the problem first before discovering the productive questions. It seems we live in a world with all kinds of things changing that were never supposed to. One of them is that we somehow have always lived in an open environment, full of uncontrolled systems behaving in quite inexplicable ways, but we could always seem to get away with only studying the things that appeared well represented with equations.

Now we can’t, and nature is giving us more than a gentle nudge toward studying them in earnest by disrupting our global economic and cultural system with them.

I’m back from Hong Kong! It was a great trip, and my talk went well—thanks to everyone here who helped out.

My talk was addressed to an audience of ‘people like me’: pure mathematicians and mathematical physicists who work at universities doing a mixture of teaching and research. I was trying to convince them to work on issues relevant to energy and the environment… and to give them some ideas of what they could do.

There were lots of questions. People seemed to find it interesting and thought-provoking. The only big piece of skepticism was someone who doubted my figures on how much carbon is burnt by a round-trip flight from Hong Kong to San Francisco. He thought the Terrapass calculation of 1.8 tonnes of carbon, or 4038 pounds of CO_{2}, per person was too high. So, I want to do that calculation myself soon.

I’m not sure I convinced anyone to instantly drop what they were doing and help save the planet… but of course I didn’t expect to. I did get some interesting feedback, though.

For example, I met one American mathematician who said she wanted to switch from pure math to something like climate change. She told me some interesting stories of her struggles to switch. For example, she went to a climate change workshop at the Mathematical Sciences Research Institute — maybe it was this — which she considered rather unsuccessful. She said it was hard for mathematicians to talk to climate scientists for long enough to make real progress. This made me think that someday, when I learn more, I could help serve as a catalyst. I told her about the Azimuth Project, and I’ll keep in touch with her.

She also said that the NSF is funding mathematicians to work in other (non-math) departments for a year and learn other subjects. Maybe when I get back to the US I can try something like that.

Now, elsewhere on this blog streamfortyseven wrote:

Nobody works for free, not mathematicians or physicists, and definitely not lawyers. I haven’t seen anything here on Azimuth about sources of funding for research in any of these topics.

There are a few aspects here:

1) First, ‘people like me’ often have a fair amount of freedom to do research on whatever we want as long as we 1) teach, 2) publish research papers in math journals and 3) try to get grants. Depending on the country and the institution, actually getting grants may be considered essential, or merely desirable.

(In the US, for example, it’s rather hard for mathematicians to get grants, so many math departments will not be upset if you don’t, though they’d be happier if you did. I’ve been lucky to be in a department like that for most of my life. Only in the last few years have I bothered to seek grants.)

As far as I can tell, there is more funding for mathematicians to work on environmental and energy issues than on pure mathematics or mathematical physics.

So, convincing people like me to switch to more practical research may be a ‘downhill struggle’ as far as money is concerned. We’re used to explaining to administrators and governments why it’s good to think about esoteric stuff like higher gauge theory, division algebras and superstrings. If we switched to climate models or mathematical biology, that job would get easier. They’d probably say “at last, a mathematician doing something that actually matters!” And it would probably become easier to get money.

The hard part about convincing people like me to work on environmental and energy issues is showing that there exist mathematically interesting projects that actually accomplish something slightly useful. I need to work up a list of such projects. Even better will be doing some such projects myself. I’m just getting started on that, with my ‘network theory’ idea.

2) Another aspect is helping people find ways to get paid to work on energy and environmental issues. This is incredibly important for people who aren’t ‘like me’: ordinary folks who need to get paid to do X before they start doing X. You’re right, I haven’t done much to help people like this find funding or jobs. I should start. Heck, all of you should start! I guess I can catalyze this by doing a blog entry on it.

3) Another aspect is convincing institutions to fund work on energy and environmental issues. I haven’t done anything about that.

She said it was hard for mathematicians to talk to climate scientists for long enough to make real progress.

And for what reasons, specifically?

Of course climate science is based on physics, so knowing some physics is necessary in order to understand and talk about it, and many mathematicians don’t have the necessary background knowledge.

Then every community has its own language which one has to learn first, and I don’t expect that climate scientists will learn math talk, it will have to be the other way around.

Last but not least I’d expect that the math problems that are of importance to climate science come from some very specific topics, like statistics and numerical approximations to partial differential equations. A pure mathematician looking for applications of number theory or algebraic geometry will surely be lost and frustrated.

He thought the Terrapass calculation of 1.8 tonnes of carbon, or 4038 pounds of CO2, per person was too high.

Terrapass may have an economic incentive to overestimate carbon emissions, since they probably reason in terms of profit maximalisation:

We’re proud of our for-profit status.

But maybe one can just ask that they provide details of one particular calculation, e.g. for educational purposes, since they write:

We are extremely open with our customers. We value our integrity above all else, and we aren’t afraid to tackle difficult questions in a forthright manner.

Especially if your calculation would yield something different, it would be interesting to compare.

Terrapass may have an economic incentive to overestimate carbon emissions…

Superficially, yes. But they also have an economic incentive to be as accurate and transparent as possible, because lots of people are suspicious of carbon offsets, and their website is publicly accessible, so any ‘cheating’ will soon be caught.

Well, I didn’t mean pure cheating but rather that they might consistently take upper or lower values that favour their profit. For example, I wonder how full planes, trains, buses are in those calculations. Being on a private jet compared to a fully booked one could easily yield a factor of 100 difference. I guess it doesn’t make a very big difference for the fuel consumption.

Anyway, your comment about carbon vs carbon dioxide has already settled the issue about your talk.

Hmm, interesting. Your calculations match those of Terrapass pretty well.

On the other hand, I made an idiotic mistake. I wrote:

He thought the Terrapass calculation of 1.8 tonnes of carbon, or 4038 pounds of CO_{2}, per person was too high [for a roundtrip flight from Hong Kong to San Francisco].

But 4038 pounds of CO_{2} is 1.8 tonnes of CO_{2}, not 1.8 tonnes of carbon!

And, this idiotic mistake infected my talk. So, the sceptic was correct to doubt me.

He didn’t believe that (according to my mistaken claim) flying from Hong Kong to San Francisco requires burning about 900 kilograms of carbon, roughly 10 times ones own weight.

As we should all know, 1 kilogram of carbon burns to form 44/12 kilograms of CO_{2}. So, I was off by a factor of about 3.66.

So, flying from Hong Kong to San Francisco burns only 250 kilograms of carbon, roughly 2.5 times ones own weight. But that’s still rather impressive.

My talk is corrected now, and I’ll try to find that guy’s email address and thank him!

Flying is more ecologicically destructive than simply measuring the tonnes of carbon. If I burn fuel at ground level, there are plenty of processes that reduce the amount of carbon dioxide reaching the upper atmosphere where it will account for global warming (primarily, being dissolved in water or photosynthesized by plants). However planes put much of the carbon directly into the stratosphere.

The tropospheric vertical mixing is of course rapid, since there is lots of convection. Roughly distance divided by wind speed – i.e. a scale of hours or days. The lower stratosphere is quite different; I remember figures of a year or two.

Thanks! Meanwhile I’ve found that for the lower part of the troposphere (the boundary layer, up to 3 km) there’s a timescale (not precisely mixing) of about an hour.

But I am a bit puzzled how you relate vertical mixing time with horizontal wind speed:

She said it was hard for mathematicians to talk to climate scientists for long enough to make real progress.

And for what reasons, specifically?

First she attended a National Science Foundation program where climate scientists, mathematicians and economists were supposed to talk to each other and agree on recommendations about what kinds of research should be funded. I get the feeling that this program was too short for deep communication to occur. At the end they quickly decided on some recommendations and left.

Later she tried to talk to a climate scientist at her university. He said he was very busy but would be free to have a conversation in… two months! She decided that at this rate, she wouldn’t make much progress.

Nathan wrote:

What kind of climate-related mathematics was the American mathematician hoping to do?

She’s not sure yet. She said she wants to do something useful. Her expertise is in analysis: more precisely, operator theory, functional analysis, and control theory. The good news: she has a 5-year grant and wants to spend this time changing research directions.

So, the challenge seems to be finding the right person to talk to.

Does anyone here know any good applications of control theory to climate science or to anything practical related to environmental problems? Something where there are important open questions, preferably of a mathematical nature?

Does anyone here know any good applications of control theory to climate science or to anything practical related to environmental problems? Something where there are important open questions, preferably of a mathematical nature?

Two points to lower the working morale:

– I don’t think it is a wise idea to follow the line “I’m an expert in x, so where are the important applications of x?”. If you would really like to have some impact, you’ll have more success if you ask “what are the most important applications and what kind of mathematics is needed to help out?”.

– Most of pure mathematics is of no interest to practitioners. If you are a pure mathematician interested in control theory, maybe you could get interested in devising numerical approximation schemes and prove their convergence. See, for example,

– Harold J. Kushner: “Numerical Methods for Controlled Stochastic Delay Systems” (Birkhäuser 2008)

This is already pretty much “applied”, but practitioners like climate scientists will have not interest in mathematical proofs of the convergence of anything, they program away and assess if their model behaves as it should. The question of the implementation mirrors a method that converges in theory is usually of minor importance.

So be warned: Anyone working in an “applied” science like climate science will stall you the moment you start talking about “interesting” math, and for good reasons.

I hope that this kind of peeemptive frustration helps to soften the culture shock.

Kushner’s book has references to some applications in ecology and biological systems, I’m sure that it will be interesting to certain flow problems, too. If anything about controlling flows through pipe systems could be applied to climate models is an interesting question, at least from my viewpoint. Two remarks:

– the book is about nonlinear systems, which is important for all really interesting applications,

– it includes stochastic influences, which makes the models richer and more realistic, think of stochastic influence as a concise model of length and time scales that cannot be resolved explicitly,

– it includes delayed influence, which means that one can include effects like long term heat transfer in the oceans to a model of the atmosphere, which is vital in climate models.

If you combine “nonlinear”, “stochastic” and “delayed”, you get models that can get pretty close to the real thing (I’m thinking of AO climate models), and you get to the current research frontier.

I agree that practitioners often have little interest in pure mathematics (but that mathematicians may nevertheless identify applied mathematics problems).

I disagree that the “All I have is a hammer, everything looks like a nail” approach is bad. If you’re an expert in control theory, why not look for applications in control theory, rather than re-train for some other application you think is more important? Aptitude for and interest in the problem count for as much as importance of the application.

I can imagine applications of control theory to questions of policy (decision making). As it happens, I attend a lot of environmental and energy policy talks here, and I’ll summarize some of what comes to mind. I don’t know if they’ve all published what they’ve talked about, but if you can’t find papers, you can Google their names and “Princeton” to find their talk abstracts.

Doug MacMynowski at Caltech is thinking about how control theory methods can be applied to geoengineering, and has looked at how transfer function approaches (not strictly control theory, but used by it) can be applied to understand ENSO dynamics. I spoke to him a bit when he visited here, and he’s very interested in thinking about how control theory can be applied to climate.

More generically, there are all kinds of applications of stochastic dynamic programming to determine, e.g., economically “optimal” emissions mitigation pathways and such. But dynamic programming is more computer science-y and algorithmic than traditional mathematical approaches to control theory.

I think a huge upcoming application of control methods could be in power systems engineering, to the renewable energy grid. Namely, managing power output and bringing backup generators and storage systems on or offline, or modulating supply and demand via the smart grid, to cope with volatile availability of solar and wind power, with uncertain demand.

However, I went to another talk by Mihai Anitescu at Argonne who persuasively argued that, in practice, one can get big gains by integrating real numerical weather forecast simulations into solar and wind power control systems, instead of treating the weather as a simple random variable. For that you’re back to computational approaches and probably dynamic programming.

Panos Parpas at MIT gave a talk on applying reduced-order modeling, multiscale simulation, and Markov processes to develop energy system control algorithms.

Control theory applies to the economics electricity markets as well. For example, see “A control theorist’s perspective on dynamic competitive equilibria in electricity markets” by Sean Meyn of UIUC. He gave an interesting talk here on how he thinks traditional approaches to real time trading are ineffective or erroneous.

One could imagine many control theory applications to ecosystem management, but I don’t know much about that. Googling ecosystem management and control theory turns up some papers.

I can’t speak to whether any of these applications are good mathematics. But I’m sure you can find at least some good mathematics related to environmental or energy control theory.

You need the word 'latex' right after the first dollar sign, and it needs a space after it. Double dollar signs don't work, and other limitations apply, some described here. You can't preview comments here, but I'm happy to fix errors.

John asked:

There are a lot of historical examples, of course, and I doubt that there are examples where some mathematical discovery had some impact directly, without the usual time span it takes to figure out what to do with it…

What about wavelets?

Wavelets were developed to analyze time series in geosciences, as a means to separate effects with long range correlations that act on widely different length and time scales. It took some time until mathematicians discovered that they were doing quite similar things in harmonic analysis. The topic evolved around a close interdisciplinary collaboration, highlights include

* on the math side: the discovery of the Deaubechie wavelets and other functions that introduced new “special” functions into mathematics on par with the Bessel functions etc.,

* on the application side new algorithms for the analysis of wavelets and the compression of data, inluding the file formats JPEG for images and MP3 for sound, for example.

How about applications of wavelets to weather prediction or climate prediction? Do you know some interesting work along those lines?

No, but of course there are examples where weather and climate data time series both from observations and from climate models have been analyzed with wavelets.

Here is an educated guess: spectral methods seem to be popular for the global solution of (an appropriate approximate version of) the Navier-Stokes equations for the atmosphere. A somewhat older reference for this kind of work is

* T.N. Krishnamurti, H.S. Bedi, V. Hardiker, Leela Watson-Ramaswamy: “An Introduction to Global Spectral Modeling” (Springer, 2nd edition 2010)

Spectral methods are about approximation in certain function spaces, for global approximations one uses of course sperical harmonics. Now there are various reasons that come to my mind why wavelets aren’t used here (unless they are, of course):

1. people think it is to hard to implement a method using wavelets,

2. people did not think about this before, because wavelets are considerably younger than the code snippets used to simulate flow dynamics,

3. people are satisfied with the existing solution and concentrate on other problems.

Actually I think it is a combination of all three. It’s the points 1 and 2 where mathematicians with some affinity to programming could help. IMHO it would be worthwhile to think about climate models with adaptive methods, that increase and decrease the local model resolution within the bounds of what is computationally feasable, depending on a metric that measures if something interesting is happening somewhere (formation of a storm, say). Of course I would be very glad if someone who knows more about this stuff and reads this would care to comment.

There is one paper about large eddy simulations (LES) of fluid flows that I found about wavelets:

– Marta de la Llave Plata and Stewart Cant: “On the Application of Wavelets to LES Sub-grid Modelling” (in Pierre Sagaut, Bernard J. Geurts, Johan Meyers (editors): Quality and Reliability of Large-Eddy Simulations, Springer 2008)

Naively, there does not seem to be a reason why this cannot not be useful for climate modeling.

I’ve never seen an academic paper on this, and it’s arguably more algorithmics than maths, but in terms of (minor) real world effects, theres UPS software planning routes to avoid left turns, which cuts engine idling time and hence CO2 emissions.

Left turns being the American and European version of right turns for Britain and Australia, and presumably other countries…:) I was a little confused for a moment as to why one’s engine would idle in turning left, but then it struck me.

I (almost) love your last sentence:

I would say “should”: If mankind still has a grain of scientific (as opposed to technological) pride like in gone centuries, we should at least try hard to understand what we’ve ruined. Anyhow it is beautiful stuff, and beauty is worth study on it’s own right. But we can also expect that a deeper understanding will have practical benefits. (Just like General Relativity etc. has meanwhile very practical applications in my mobile navigation system.) Moreover I expect deeper philosophical enlightenment: The old world of math and physics was a world made of dead matter. Now we should care about living matter. E.g. to make Lovelock’s Gaia theory rigorous (perhaps deduce it from thermodynamics) would be a great milestone in the history of natural philosophy.

The old world of math and physics was a world made of dead matter. Now we should care about living matter.Good line. Also, the old world was concentrated (energy) and the new world is diffuse (entropy). Most likely that many near-term forms of energy are based on dispersed phenomena such as wind, solar, bio, etc.

Florifulgurator wrote:

Okay, now it says “should”. I forgot that in this section I’m boldly telling mathematicians what they

shoulddo, not predicting what theymaydo!Meanwhile I feel a bit embarrassed for boldly telling you (grandmaster Baez) what you “should” tell … :-)

In fact I may change it to “will”. When you say “you should study X”, it makes academics want to fight back. But if you say “everyone will be studying X”, it’s more likely to make them want to go along. Especially if it’s true.

Fori, Well, that’s assuming that the biosphere is generally like an equation… Net-energy systems might be energetic processes of complex organizational accumulation though, often having parts that are actively learning as they go, rather than equations.

If so, then to find equations to help work with them you’d need to study something *other than* how to think of them as equations… Rosen and others have considered it simply as having two subjects. One is the natural world that behaves by itself that we study. Then the other is the world of our theories we tinker with, looking for what will help us interact with the one.

Phil, I shouldn’t go out on this limb in the presence of real math masters… Still: Methinks math is more than about equations. Perhaps we can model the interknit learning phenomena with n-categorical meta math, plug that into an infinite dimensional information geometric space (made of measures representing the spatiotemporal distribution of living matter), etc., and get some qualitative information of how things behave. Or get a general recipe for model numerics (perhaps like the Runge-Kutta/renormalization connection, cf. Butcher group for a taste of such math).

The perspective has been changing from “equations that have to be solved” to “maps of objects whose properties we have to study” in mathematics, you can see this if you follow the development of functional analysis from around 1900 (it is about solving differential and integral equations) to today. Today we don’t study solutions of Fredholm integral equations, but the properties of Fredholm operators instead, for example.

So it’s no longer about equations :-)

Flori, Well, yes that describes a strategy, and could be used with an object language for components like Tim suggests. BUT even if that were feasible and not just poetic, aren’t you still stuck in the same situation of having to learn how to understand complex systems from observing their behavior, and not from having programmed it?

Isn’t that the step everyone seems to be avoiding, learning how to understand complex systems from watching how they work rather than from knowing the rules they follow?

Phil, I try to follow what you’re saying, but it just isn’t clear. Your website is also muddled. I suppose the opinion you are expressing is just “a theory of how water flows is futile because we can never understand the precise structure which dictates the lifecycle of water particles”. (And at bottom, isn’t that precisely what you are suggesting?). Before we give any part of ourselves to a field, we first ask whether it might be interesting, and here our ignorance and prejudice is valued, here we decide whether we shall proceed or move on to another. I mean, we love films, say, and by what other process do we ever come to watch a film? I certainly have my own opinion–based on nothing but ignorance and taste–that meteorology (here i take the naive view that the business of meteorology is prediction) is bounded. That is, I’m amused by forecasters who credit the limitation of weather predictions to inadequacies in our models.

But on all this i’m an amateur and mildly embarrassed to fill the blog with such low quality ramblings. I would hope my comments are deleted–in the name of waste management.

Basically, I don’t understand what you’re saying. I’m under the impression that you think mathematicians believe themselves to possess some view of ”truth” in that ridiculous sense of old french philosophers.

Justin Martel wrote:

Weather models model the atmosphere only, they basically monitor what get’s blown our way. But that still means that the horizontal grid resolution is around 5 km, which is not fine enough to resolve many cloud formation processes, for example. From the viewpoint of a single human on the surface, to have one grid point every kilometer isn’t impressive, right? But these are the limits due to limited computational resources. When you set up a global model, a quick back on the envelope calculation can tell you for example that combining a 50 km horizontal grid spacing with 25 vertical grid cells leaves you with 5 kB of memory per grid cell if you have 1 GB memory for grid data overall.

Simply put: There are a lot of relevant physical processes that cannot be resolved in weather and climate models, but have to be put in by hand via heuristic corrections, due to the coarse grid that is enforced by limited computer resources.

Phil, I’m not sure I understand your objection: You seem to say we can’t in principle understand much (if anything) of the workings of nontrivial systems – because that would require us to predict their inventions (reorganizing micro details, inventing new macro rules).

Indeed methinks that would require us to be all-knowing Laplace demons. It would be like wanting to predict detailed evolution of life forms.

(Who is “Rosen and others”? Quote/link?)

But I think we can understand qualitatively and partially how complex systems work. Perhaps the success of physics is an example for that: Nature is full of emergent properties where the exact micro details don’t matter and some “higher principle” is responsible, like maximum entropy production. Some emergent properties can be modelled mathematically – and that’s probably the reason for the unreasonable effectiveness of mathematical equations. That we know some fundamental constants with great precision is perhaps due to them describing emergent properties. Of course, observing and learning about complex system behaviour is an essential part of natural philosophy. There is perhaps not much difference between “dead matter” and living matter.

Here’s another example of “qualitative math”: Estimating the topology of spaces from their curvature (e.g. Betti numbers). Earth is finite because its curvature at any point is positive.

Mr. van Beek, suppose you had unlimited computational power ie. you could instantaneously measure every square meter of this planet. That is, suppose you had all the information you could possibly ever want. Would that not paralyze you?

Aren’t our models essentially based on the fact that we have limited and coarse input data?

Permit me to be vulgar here: what is the category within which meteorologists work? To rephrase, how would one be a weatherman for 4-manifolds?

Flori, Justin & Tim, sorry I didn’t notice your questions for a couple days. I’m not in the least “criticizing” models for being naturally incomplete, but suggesting we use that property to help expose and study the ways nature does more than we can model, studying how nature completes natural system behaviors by comparing model and observed behavior. I’m suggesting a forensic science approach, comparing standard models we understand well to help reveal coherent behaviors of natural systems that diverge in the model we understand.

No doubt that seems a muddleheaded way to construct a model, as it’s something of the opposite, not how to construct a model but how to deconstruct our observations of nature. Of course, the main “coherent departure” of nature from what we can model is the emergence of new systems.

There might be other clues to use, but my theorem of emergence and continuity proves that to satisfy energy conservation new energy using processes need to exhibit a period of “inflation”. It’s a conclusion that is quite broad, that the big bang’s period of inflation, found necessary to explain the astrophysical data, would be a special case of.

Now do you understand why I’d scan data looking for emerging processes of inflationary development?

Justin Martel wrote:

You may call me Tim here, I don’t insinst on a formal introduction :-)

Computational power and better input data are of course two separate topics, to improve weather models we need to improve both. The question how far we could go in principle is of course an interesting one, but when we take a look at weather models

todayfrom apracticalviewpoint, it should be clear that there are a lot of obvious flaws and problems that can be solved/improved simply by more computational power, and more data. There are projects underway in both directions. As a weather modeller I’d like to incorporate lakes and cloud forming processes due to lakes into my model, for example. With a grid spacing of 5 km that is hardly possible.I’d really like to know long a global weather forecast could be done that is of use (being deliberatly imprecise with this question). Today the longest model runs encompass ca. 16 days, I think.

I’m sorry, I don’t understand what “category” means in this context, nor what is vulgar about the question. To a first approximation global weather modelling is computational fluid dynamics on a rotating sphere, is that your point? I suppose that this becomes a lot more complicated in 4D…

For a decade and more my wife complained about word processing people, and I kept telling her: you’ll never get documents created as you want in a timely fashion unless you do it yourself. Programming is a bit like that too: and spreadsheet programming shows that people want to program if the environment is accessible enough. It is nice when sophisticated mathematics makes a difference. It is nice when real mathematicians can cooperate with people in other fields and use semi-sophisticated mathematics to help them. Yet I feel that the real need is for everybody to understand unsophisticated mathematics enough so that when they have a problem amenable to mathematical treatment they at least recognise that and look for help. The Internet seems a wonderful tool for educating people about mathematics, but I feel there won’t be much progress until the educational authorities stop regarding mathematics as an optional skill. At any rate one of the issues seems to be that mathematicians love to generalize, but the implications of the general theorems don’t percolate down. I remember an memoir by a famous mathematician (perhaps Arnold) complaining about papers published on how to solve particular types of PDE, when they were just special cases of a “well known” general theorem. Unless mathematicians are also involved in the real world, or at least the less unreal world, then they won’t know how their understanding can make the difference it should.

If plants keep their leaf pores smaller, that means that particular plant types can grow in drier conditions. At the edge (if rainfall stays the same) that means more tree covered areas which is a negative feedback on CO2. In other places I presume it means that you eventually get different trees (basically because trees which are adapted to relatively dry conditions lose out to trees that don’t have those adaptations where those adaptations aren’t necessary). So it’s tricky. And isn’t this typical of so much relevant stuff. You see a summary of a result and you wonder: “did they allow for this or that?”. It would be interesting to develop an oracle (like Watson?) which could absorb a lot of information and answer questions like that.

There’s a ton of stuff out there on climate and wavelet modelling, just do a google search on the words…

Here:

http://motls.blogspot.com/2009/05/climate-and-morlet-wavelet-transform.html

And

http://scienceweek.com/2005/sw050325-6.htm

And

http://ieeexplore.ieee.org/xpl/freeabs_all.jsp?arnumber=5290751

And

http://www.clim-past-discuss.net/1/193/2005/cpd-1-193-2005-print.pdf

Off to dinner, now…

Thanks for the links, these are all examples of time series analysis or synthesis, for which wavelets were developed in the first place. A wavelet match of 20th century data based on some model for a random process could be used as a null hypothesis to test climate models, for example. In this sense it would be possible to use wavelets to synthesize time series that predict weather or climate data.

But when John asked

I understood the question in a different sense, however, namely if wavelets are used as a tool in climate or weather models. Wavelets were invented to replace trigonometric functions, because people needed base functions that are localized both in time and frequency. Therefore I was speculating if wavelets could be used in global climate models to replace spherical harmonics in a spectral approximation, because then it would be possible to adjust the resolution of the approximation locally.

This is much like hp-FEM, or, to be more precise, hp-FEMs are an example of this method with a very simple wavelet basis (or, rather, a frame) consisting of localized polynomials.

Spectral methods are very popular because they allow the explicit control of the wave modes that are resolved in a model, which is precisely what you want from a climate model: Retain only the large scale effects. It would be interesting to know if you get different results when you enable the model to change its resolution locally adaptiveley.

Ok, try this:

http://www2.iwr.uni-heidelberg.de/groups/CoVis/Data/Papers/vis09.pdf

And maybe this:

http://adsabs.harvard.edu/abs/2005AGUFM.H13G1392K

There might be something of interest here, or the prof could have some useful insights:

http://people.maths.ox.ac.uk/moroz/

Maybe this:

http://www.citeulike.org/user/nurban/article/3803391

And if none of this is what you’re looking for, then you can find your answer as to who is going to figure this out by looking in a mirror…

There is one project mentioned on the page of Dr Irene M. Moroz which is close to what we were discussing:

Yes, that’s what I head in mind. I recently “discovered” Burgers’ equation as a simple model to start with myself, and have written a little bit about it here on the Azimuth wiki. Writing a Galerkin = spectral approximation for this equation is already a formidable homework.

Tim wrote:

Yeah, that’s sort of what I had in mind: using wavelets to describe fluid flow, as part of simulating the weather. The paper you mentioned about large eddy simulation sounds like it’s heading in that direction, though I haven’t read it.

The problem is that I don’t know for sure either, especially if LES in the field of computational fluid dynamics is the same as LES in meteorology. The paper is about CFD, not about meteorology.

I forgot to mention that the climate and weather modeling books that I have mention “LES” as the most commonly used approximation technique for weather and climate models, but I’m not sure if they have the same understanding of what LES is as the CFD people who concentrate on engineering applications of fluid flow.

Azimuth saw some crowd-sourcing for a talk…

My example is not of a result but of an area of mathematics, namely random trees and branching processes. There are lots of examples in biology. The tree of life is important for understanding and evaluating biodiversity. Your body contains about 10

^{}^{14}cells, each one produced by binary fission from a parent, back to a single zygote: that is quite a big tree. It contains sone interesting subtrees, such as the airways of your lungs, and models of those are useful for understanding the effects of air pollution for example. And the woody kind of tree, and their leaves can also be modelled by branching processes. From physics, there is the shower of particles (up to a billion) that can be produced when a cosmic ray hits the atmosphere – something which may have an important role in cloud formation.Two search terms:

Bellman-Harris branching processes

Crump-Mode-Jagers branching processes

Two more recent practitioners:

David Aldous (http://www.stat.berkeley.edu/~aldous/)

Mike Steel (http://www.math.canterbury.ac.nz/~m.steel/)

Two relevant articles:

• Bernd Sturmfels, Can biology lead to new theorems?

• J.E. Cohen: Mathematics is biology’s next microscope, only better; biology is mathematics’ next physics, only better,

PLOS Biology2(2004) No.12.Thanks, Graham! This paper is great fun:

• Bernd Sturmfels, Can biology lead to new theorems?

A nice quote at the start, something Rota said before 1986:

Sturmfels’ work applying toric varieties to various areas in biology and chemistry is something I want to explain when I get back from Hong Kong and start talking about stochastic Petri nets.

Rota’s question of why Maximum Entropy Principle works as well as it does.

As Tim suggested, there are a number of impressive, directly impacting developments in time series analysis. But wavelets, although nice, aren’t actually exciting – just special cases of spectral analysis.

I think the most exciting result in time series analysis was the discovery of universality in chaos (Feigenbaum et al.). I don’t know about the direct impacts, however.

I don’t know of any practical applications of chaos theory/ fractals, either. Wavelets, especially the discrete fast wavelet transform – went from the blackboard directly to the assembly line, that’s why I mentioned it.

Of course every physicist knows a lot of examples of mathematical discoveries that became important for applications – but usually much, much later.

John F wrote:

I guess we all get excited by different things, but back when wavelets first came out, I seem to recall a bunch of mathematicians being excited. If you google wavelets excitement you’ll get some interesting hits.

Here’s an intro for mathematicians, back from those days I guess:

• James S. Walker, Fourier analysis and wavelet analysis,

AMS Notices44(1997), 658-670.It doesn’t capture a sense of ‘excitement’, but it’s nice and clear, at least for mathematicians who are already comfortable with Fourier analysis and want to learn about wavelets.

However, this article has nothing to do with energy, the environment, biology, etc.

Computers of course. The first televised weather program was shown here in Sweden first in september 1954 with Rossby and the BESK computer and then John von Neumann and R Charney did a televised forecast six months later using “his” ENIAC.

And the second must be topology, which has given us means to understand and predict nonlinear chaotic phenomena which we hopefully can be able to control as well. (Source: Eugenia Kalnay,

Atmospheric Modeling, CUP 2003.)Just a couple of quick observations about, sensors, signal processing, and climate change. Signal processing is at the heart of detecting “hidden” signals in noise, signals the are slowly varying in time, and anomalous behavior detection of phenomena using sensors. Wavelets have become increasingly important for different sensor applications including solving of PDE’s related to weather modeling and forecasting. Also, see

wavelet + radar + weather for detecting anomalous weather such as turbulence, tornadoes, etc. Wavelet have been widely used in ground penetrating radar, which have been used for detecting buried objects but can also be used detecting changes in soil characteristics for monitoring how climate zones change.

Another example is time-frequency signal processing methods which were pioneered by Leon Cohen of Hunter college in the paper: “Generalized phase-space distribution functions” – Journal of Mathematical Physics, 1966. This paper was considered foundational to signal processing work on time-frequency methods for signal processing that revolutionized examination of signals that change with time and recognizing them. While this is widely used in radar, sonar, and acoustical data analysis, it has found limited use in climate change per say. Climate data presented this way would probability make some aspects of change much more apparent as they have in other time varying signals generated by nature and detected by sensors.

This would be fun to do with as a student thesis project for those of you looking for ideas for student projects.

Ensemble Kalman Filters seem to be used a lot for data assimilation in weather forecasting, not sure if and how this translated to climate prediction, but here are a few links:

http://arxiv.org/abs/0712.3965v2

http://nldr.library.ucar.edu/repository/assets/osgc/image_201009.pdf

http://enkf.nersc.no/Publications/eve09a.pdf

I have actually read them and i especially liked the first one about wildfires.

There is some work on Terrestrial Hydrology:

http://iopscience.iop.org/1742-6596/22/1/010/pdf/1742-6596_22_1_010.pdf

But I have seen more. Especially a lot of work in the systems biology area recently. Not to mention modeling and control of large scale electrical networks, variable speed wind turbines, and of course, Tokamak plasmas. They all have a potentially big impact on environmental issues.

If anyone is interested in one area in particular i can try to dig out more literature on Mathematical Modeling and Control in that specific area.

Ensemble Kalman filters (EnKF) have been used for parameter estimation problems in climate modeling. They are now gaining popularity for state estimation in decadal prediction problems, for the same reasons they are used in weather prediction: sensitivity of the system state to initial conditions.

Here are a few standard references to wavelets and signal processing.

Stephane Mallat,

A Wavelet Tour of Signal Processing. Look at both the second and third editions which cover somewhat different material.Jean-Luc Starck, Fionn Murtagh, and Jalal M. Fadili,

Sparse Image and Signal Processing: Wavelets, Curvelets, Morphological Diversity, Cambridge 2010. Many up-to-date references.For sheer beauty, check out http://www.curvelet.org/.

Here is a paper that I feel is of great importance to climate studies:

• David Donoho, Arian Maleki, Inam Rahman, Merteza Shahram, and Victoria Stodden, 15 Years of Reproducible Research in Computational Harmonic Analysis.

The abstract is:

There is a reproducible research webpage at http://www.rrplanet.com/ .

That’s a very interesting article—thanks! I think Tim van Beek will be interested in it. I like this quote (a quote of someone in the article, not a quote of the article itself):

I think the world-wide web is a prerequisite for being able to publish the actual scholarship. So before the web existed, there was an excuse for not doing this.

The title of Sturmfel’s paper is provocative and seriously disappointing. It seems very possible that mathematics is biologically impotent.

At bottom, I think our business as mathematicians is simple: we solve problems. We are attracted to question marks and contradictions. If biology is to attract mathematicians, then she must lure us with problems. And something beyond the usual, and tedious, and dull, and wholly predictable and tiring affairs of dynamical systems and population growths or of modelling weather. And if she has nothing to lure the young and intrepid, what then?

Biology is

fullof question marks and contradictions. It’s physics, not biology, that studies the “dynamical systems” you find tedious, dull, wholly predictable and tiring. Biology demands new ideas that we’re just beginning to have.(By the way: if you think the subject of weather is dull, you either haven’t studied it very much, or your tastes are just fundamentally different than mine. The more I learn about it, the more interesting it gets.)

I misunderstood the main interest of this blog. The question of “mathematics and the environment” is not that of the relation between “mathematics and biology”. As a student not exactly interested in fluid dynamics (which i consider–without any real experience or justification–as the actual domain of meteorology and weather) i see here nothing compelling.

Sure, among experts, the questions are enthralling. But does it have anything to catch the eyes of those whom it has not already absorbed? I mean, something beyond wavelets. It seems to me that is the first problem of the whole field—it’s closed and hides itself from mathematicians. Does it find no stronger ally in topology than the statement that somewhere the wind is not blowing, or that somewhere barometric pressure and “something else” coincide?

I’m obviously ignorant, prejudiced, and skeptical. But I’m also a young energetic student who’s looking for something fascinating. And when I turn my way towards, for instance, this website and the comments, I find only the typical talk of applied mathematicians. I conjecture that a field is only significant if it is interesting geometrically. And so?

One may either dismiss everything i’m trying to say as the ramblings of an obnoxious kid, or one may try to catch the song of a rare bird flying around this field, and playing it in the ears of those passerby’s who are looking around, trying to find something to tap their foot to.

Note: if the interest of this blog was actually biological, I would begin by referring the readers to the article by Gromov and Carbone titled “Mathematical Slices of Molecular Biology”.

Justin wrote:

I’m no expert. I may however find it easier than you to become interested in things. I often wish I had days that were twice as long, and twice as much physical energy, to learn about all the fascinating things in this world. I have a bunch of books on my desk:

Fundamentals of Chemical Kinetics, andThe Theoretical Biologist’s Toolbox, andWetlands Ecology: Principles and Conservation, andChemical Reaction Networks: A Graph-Theoretic Approach. They’re all really cool, but unfortunately I don’t have time to read them all, because I want to keep explaining network theory and how stochastic Petri nets can be understood as a variant of quantum field theory in which probabilities replace amplitudes. Since stochastic Petri nets are also related to toric varieties, there should be some fun relation between quantum field theory, or at least this variant, and toric geometry. And I want to figure out what it is!I’m interested in that too! Thanks!

What I’m pointing out is that inventing your own models of natural systems to study, as physics relies on, has major drawbacks. It represents nature as designed according to your own modeling technique and value system, that you embed in constructing your model. So you’re not studying your subject in its natural form at all, and having no way to discuss what the significant difference between nature and model might be. You only get to discuss the significance of differences between model A and model B.

It results in your only studying imaginary systems, is the simple way to state the basic dilemma. It means losing all interest in how natural systems have fluidly changing organization and widely distributed animating parts, causing you to fail to look for or find the questions about such things that actually are answerable. Studying only imaginary systems diverts interest from the study of the natural processes themselves.

There are an enormous range of things that artificial systems can’t represent that are quite common behaviors and organizational states found in uncontrolled natural systems in open environments. That appears to be a central reason for why modeling has not been of much help for understanding them.

Granted, you need to look at the problem first before discovering the productive questions. It seems we live in a world with all kinds of things changing that were never supposed to. One of them is that we somehow have always lived in an open environment, full of uncontrolled systems behaving in quite inexplicable ways, but we could always seem to get away with only studying the things that appeared well represented with equations.

Now we can’t, and nature is giving us more than a gentle nudge toward studying them in earnest by disrupting our global economic and cultural system with them.

I’m back from Hong Kong! It was a great trip, and my talk went well—thanks to everyone here who helped out.

My talk was addressed to an audience of ‘people like me’: pure mathematicians and mathematical physicists who work at universities doing a mixture of teaching and research. I was trying to convince them to work on issues relevant to energy and the environment… and to give them some ideas of what they could do.

There were lots of questions. People seemed to find it interesting and thought-provoking. The only big piece of skepticism was someone who doubted my figures on how much carbon is burnt by a round-trip flight from Hong Kong to San Francisco. He thought the Terrapass calculation of 1.8 tonnes of carbon, or 4038 pounds of CO

_{2}, per person was too high. So, I want to do that calculation myself soon.I’m not sure I convinced anyone to instantly drop what they were doing and help save the planet… but of course I didn’t expect to. I did get some interesting feedback, though.

For example, I met one American mathematician who said she wanted to switch from pure math to something like climate change. She told me some interesting stories of her struggles to switch. For example, she went to a climate change workshop at the Mathematical Sciences Research Institute — maybe it was this — which she considered rather unsuccessful. She said it was hard for mathematicians to talk to climate scientists for long enough to make real progress. This made me think that someday, when I learn more, I could help serve as a catalyst. I told her about the Azimuth Project, and I’ll keep in touch with her.

She also said that the NSF is funding mathematicians to work in other (non-math) departments for a year and learn other subjects. Maybe when I get back to the US I can try something like that.

Someone told me I should talk to the head of the Institute for Mathematical Sciences here at NUS—they do a lot of interdisciplinary stuff. I’ll start by going to their workshop on Probability and Discrete Mathematics in Mathematical Biology. Unfortunately I missed some good talks while in Hong Kong! But it’s not over yet.

Now, elsewhere on this blog streamfortyseven wrote:

There are a few aspects here:

1) First, ‘people like me’ often have a fair amount of freedom to do research on whatever we want as long as we 1) teach, 2) publish research papers in math journals and 3) try to get grants. Depending on the country and the institution, actually

gettinggrants may be considered essential, or merely desirable.(In the US, for example, it’s rather hard for mathematicians to get grants, so many math departments will not be upset if you don’t, though they’d be happier if you did. I’ve been lucky to be in a department like that for most of my life. Only in the last few years have I bothered to seek grants.)

As far as I can tell, there is

morefunding for mathematicians to work on environmental and energy issues than on pure mathematics or mathematical physics.So, convincing people like me to switch to more practical research may be a

‘downhill struggle’as far as money is concerned. We’re used to explaining to administrators and governments why it’s good to think about esoteric stuff like higher gauge theory, division algebras and superstrings. If we switched to climate models or mathematical biology, that job would get easier. They’d probably say “at last, a mathematician doing something that actually matters!” And it would probably become easier to get money.The hard part about convincing people like me to work on environmental and energy issues is showing that there exist

mathematically interesting projects that actually accomplish something slightly useful. I need to work up a list of such projects. Even better will be doing some such projects myself. I’m just getting started on that, with my ‘network theory’ idea.2) Another aspect is helping people find ways to get paid to work on energy and environmental issues. This is incredibly important for people who aren’t ‘like me’: ordinary folks who need to get paid to do X before they start doing X. You’re right, I haven’t done much to help people like this find funding or jobs. I should start. Heck,

all of youshould start! I guess I can catalyze this by doing a blog entry on it.3) Another aspect is convincing institutions to fund work on energy and environmental issues. I haven’t done anything about that.

John said:

Glad to hear that!

And for what reasons, specifically?

Of course climate science is based on physics, so knowing some physics is necessary in order to understand and talk about it, and many mathematicians don’t have the necessary background knowledge.

Then every community has its own language which one has to learn first, and I don’t expect that climate scientists will learn math talk, it will have to be the other way around.

Last but not least I’d expect that the math problems that are of importance to climate science come from some very specific topics, like statistics and numerical approximations to partial differential equations. A pure mathematician looking for applications of number theory or algebraic geometry will surely be lost and frustrated.

Good to hear your talk went well.

John wrote about someone:

Terrapass may have an economic incentive to overestimate carbon emissions, since they probably reason in terms of profit maximalisation:

But maybe one can just ask that they provide details of one particular calculation, e.g. for educational purposes, since they write:

Especially if your calculation would yield something different, it would be interesting to compare.

Frederik wrote:

Superficially, yes. But they also have an economic incentive to be as accurate and transparent as possible, because lots of people are suspicious of carbon offsets, and their website is publicly accessible, so any ‘cheating’ will soon be caught.

Well, I didn’t mean pure cheating but rather that they might consistently take upper or lower values that favour their profit. For example, I wonder how full planes, trains, buses are in those calculations. Being on a private jet compared to a fully booked one could easily yield a factor of 100 difference. I guess it doesn’t make a very big difference for the fuel consumption.

Anyway, your comment about carbon vs carbon dioxide has already settled the issue about your talk.

What kind of climate-related mathematics was the American mathematician hoping to do?

I decided to check the flight CO2 emissions using some numbers I googled off the web.

Assumptions:

11000 km trip

3.5 liters of jet fuel used, per 100 km, per passenger

0.8 kg/liter jet fuel density

3.1 kg CO2 (or CO2eq?) / kg jet fuel

Result: 955 kg CO2/passenger for a Hong Kong->San Francisco flight, or about 1 ton

Hmm, interesting. Your calculations match those of Terrapass pretty well.

On the other hand, I made an idiotic mistake. I wrote:

But 4038 pounds of CO

_{2}is 1.8 tonnes of CO_{2}, not 1.8 tonnes of carbon!And, this idiotic mistake infected my talk. So, the sceptic was correct to doubt me.

He didn’t believe that (according to my mistaken claim) flying from Hong Kong to San Francisco requires burning about 900 kilograms of carbon, roughly 10 times ones own weight.

As we should all know, 1 kilogram of carbon burns to form 44/12 kilograms of CO

_{2}. So, I was off by a factor of about 3.66.So, flying from Hong Kong to San Francisco burns only 250 kilograms of carbon, roughly 2.5 times ones own weight. But that’s still rather impressive.

My talk is corrected now, and I’ll try to find that guy’s email address and thank him!

Flying is more ecologicically destructive than simply measuring the tonnes of carbon. If I burn fuel at ground level, there are plenty of processes that reduce the amount of carbon dioxide reaching the upper atmosphere where it will account for global warming (primarily, being dissolved in water or photosynthesized by plants). However planes put much of the carbon directly into the stratosphere.

Roger Witte said:

I suppose your comment is true for short timescales because the carbon dioxide will get mixed, eventually.

Does anyone know the “vertical” mixing time of carbon dioxide in the atmosphere?

I’ve googled a little bit and found

which is not what I want, but in John’s carbon dioxide puzzles there is

The tropospheric vertical mixing is of course rapid, since there is lots of convection. Roughly distance divided by wind speed – i.e. a scale of hours or days. The lower stratosphere is quite different; I remember figures of a year or two.

John Furey said:

and

Thanks! Meanwhile I’ve found that for the lower part of the troposphere (the boundary layer, up to 3 km) there’s a timescale (not precisely mixing) of about an hour.

But I am a bit puzzled how you relate vertical mixing time with horizontal wind speed:

Tim wrote:

First she attended a National Science Foundation program where climate scientists, mathematicians and economists were supposed to talk to each other and agree on recommendations about what kinds of research should be funded. I get the feeling that this program was too short for deep communication to occur. At the end they quickly decided on some recommendations and left.

Later she tried to talk to a climate scientist at her university. He said he was very busy but would be free to have a conversation in…

two months!She decided that at this rate, she wouldn’t make much progress.Nathan wrote:

She’s not sure yet. She said she wants to do something

useful. Her expertise is in analysis: more precisely, operator theory, functional analysis, and control theory.The good news:she has a 5-year grant and wants to spend this time changing research directions.So, the challenge seems to be finding the right person to talk to.

Does anyone here know any good applications of control theory to climate science or to

anything practical related to environmental problems?Something where there are important open questions, preferably of a mathematical nature?John asked:

Two points to lower the working morale:

– I don’t think it is a wise idea to follow the line “I’m an expert in x, so where are the important applications of x?”. If you would really like to have some impact, you’ll have more success if you ask “what are the most important applications and what kind of mathematics is needed to help out?”.

– Most of pure mathematics is of no interest to practitioners. If you are a pure mathematician interested in control theory, maybe you could get interested in devising numerical approximation schemes and prove their convergence. See, for example,

– Harold J. Kushner: “Numerical Methods for Controlled Stochastic Delay Systems” (Birkhäuser 2008)

This is already pretty much “applied”, but practitioners like climate scientists will have not interest in mathematical proofs of the convergence of anything, they program away and assess if their model behaves as it should. The question of the implementation mirrors a method that converges in theory is usually of minor importance.

So be warned: Anyone working in an “applied” science like climate science will stall you the moment you start talking about “interesting” math, and for good reasons.

I hope that this kind of peeemptive frustration helps to soften the culture shock.

Kushner’s book has references to some applications in ecology and biological systems, I’m sure that it will be interesting to certain flow problems, too. If anything about controlling flows through pipe systems could be applied to climate models is an interesting question, at least from my viewpoint. Two remarks:

– the book is about nonlinear systems, which is important for all really interesting applications,

– it includes stochastic influences, which makes the models richer and more realistic, think of stochastic influence as a concise model of length and time scales that cannot be resolved explicitly,

– it includes delayed influence, which means that one can include effects like long term heat transfer in the oceans to a model of the atmosphere, which is vital in climate models.

If you combine “nonlinear”, “stochastic” and “delayed”, you get models that can get pretty close to the real thing (I’m thinking of AO climate models), and you get to the current research frontier.

Tim,

I agree that practitioners often have little interest in pure mathematics (but that mathematicians may nevertheless identify applied mathematics problems).

I disagree that the “All I have is a hammer, everything looks like a nail” approach is bad. If you’re an expert in control theory, why not look for applications in control theory, rather than re-train for some other application you think is more important? Aptitude for and interest in the problem count for as much as importance of the application.

I can imagine applications of control theory to questions of policy (decision making). As it happens, I attend a lot of environmental and energy policy talks here, and I’ll summarize some of what comes to mind. I don’t know if they’ve all published what they’ve talked about, but if you can’t find papers, you can Google their names and “Princeton” to find their talk abstracts.

Doug MacMynowski at Caltech is thinking about how control theory methods can be applied to geoengineering, and has looked at how transfer function approaches (not strictly control theory, but used by it) can be applied to understand ENSO dynamics. I spoke to him a bit when he visited here, and he’s very interested in thinking about how control theory can be applied to climate.

More generically, there are all kinds of applications of stochastic dynamic programming to determine, e.g., economically “optimal” emissions mitigation pathways and such. But dynamic programming is more computer science-y and algorithmic than traditional mathematical approaches to control theory.

I think a huge upcoming application of control methods could be in power systems engineering, to the renewable energy grid. Namely, managing power output and bringing backup generators and storage systems on or offline, or modulating supply and demand via the smart grid, to cope with volatile availability of solar and wind power, with uncertain demand.

However, I went to another talk by Mihai Anitescu at Argonne who persuasively argued that, in practice, one can get big gains by integrating real numerical weather forecast simulations into solar and wind power control systems, instead of treating the weather as a simple random variable. For that you’re back to computational approaches and probably dynamic programming.

Panos Parpas at MIT gave a talk on applying reduced-order modeling, multiscale simulation, and Markov processes to develop energy system control algorithms.

Control theory applies to the economics electricity markets as well. For example, see “A control theorist’s perspective on dynamic competitive equilibria in electricity markets” by Sean Meyn of UIUC. He gave an interesting talk here on how he thinks traditional approaches to real time trading are ineffective or erroneous.

One could imagine many control theory applications to ecosystem management, but I don’t know much about that. Googling ecosystem management and control theory turns up some papers.

I can’t speak to whether any of these applications are good mathematics. But I’m sure you can find at least some good mathematics related to environmental or energy control theory.

Modeling of adaptation has not received much attention.

http://www.nature.com/nclimate/journal/v1/n1/full/nclimate1051.html

Just came across this tutorial on control of wind turbines.

Not really advanced math, but it might still be interesting.