So much of modeling with differential equations is concerned with long-time behavior of solutions that taking time in the reverse direction is almost never seen or discussed in a Diff Eq class. Yet climate and weather modelers need to do just that - they need to model and understand the conditions of the earth’s climate in Deep Time - a time in earth’s past well beyond the current state of knowledge of earth’s climate states. They need to go ‘way back’ in time, effectively to - ? This has been pointed out with great urgency by geologist G.S. Soreghan of Univ. of Oklahoma, who writes in Lessons From Earth’s Deep Time that much is known about Earth’s climate over the last 100,000 years. This time is miniscule when compared to earth’s age of 4.5 billion years. Going back so far in time is essential in order to understand the nature and causes of the extreme climate states that existed over that vast expanse of time, climate states that, according to Soreghan, "would seem very alien" to us today. And why is this a pressing need? The answer is extremely chilling, and almost too obvious: our current path of global warming could well be a harbinger of a future earth climate that mimics some deep-time regime. Soreghan and a significant number of other concerned scientists have banded together to form GeoSystems, described as

an interdisciplinary, community-based initiative stemming from the growing recognition that a full understanding of Earth’s climate system –and our climate future– lies in examining the wealth of "alternative-Earth" climatic extremes archived in the pre-Quaternary geologic record.

Visit the GeoSystems, site for resources on deep time climate issues, including reports on extreme climate states such as the Snowball Earth scenario that is claimed to have existed over 600 million years ago. The site also lists the ongoing efforts of the concerned scientists to seek research funding (GeoScience is NSF-supported). To read more about the "alien" climates of deep-time, visit the Paleomap Project of C. R. Scotese of the Univ. of Guelph (Ontario). Here you’ll find a climate history of the earth from the Pre-Cambrian period to the present day, some fascinating animations, and a description of the methodology used to determine the deep-time climate. And you’ll also find the image of the Snowball Earth/ Mitten at the top of this post. Given the rate of increase of atmospheric greenhouse gases since the Industrial Revolution, the more appropriate image would consist of kitchen mit…and a different color earth.

Click to enlarge view of sunplumeA number of news sources reported yesterday that scientists had predicted more intense activity in the coming cycle of sun spots, solar flares, and other solar phenomena. This prediction was accompanied by another prediction on timing - the solar activity would be somewhat “delayed.“These statements naturally got me wondering about a number of issues - how is solar activity modeled? How accurate are the predictions? Is the underlying physics that causes the upsurge in activity understood?First, it’s clear why accurate predictions are needed: increased solar activity can really do a number on communication and navigation electronics, causing satellites to go awry - and in some cases causing delays of planned rocket launches.Now to the “delayed” issue - the basic lore is that the sun goes through an 11-year cycle of low-to-high activity, culminating in a significant number of sunspots appearing on the sun’s surface. In reality, there is a spread in this: cycles run from 9 to 14 years, with 11-years being the mode of the distribution of solar cycle lengths. (Solar cycles and their lenghts have been tabulated since approx. 1760. We are currently in Cycle 23, which started in May 1996. Click here for a histogram of cycle lengths .) Most solar cycles are near the 11-year length, so any delay is with respect to 11 years (i.e. the next cycle is expected to start sometime in 2008). The delay, then, is nothing out of the ordinary.Even with this spread in cycle lengths, the regularity is amazing, given the incredibly non-linear processes at work in the stellar interior - a hellish cauldron of gas and plasma, fission and fusion, in a tug-of-war between gravitational collapse and outward radiation pressure, a tussle that ends up with the solar dynamo switching its magnetic field direction every cycle.What is it that leads to this regularity? Models are of two types: empirical/statistical ones that use some serious regression techniques, and first-principles differential equation models.For a thorough description of the empirical models and a successful way to combine the positive features of these models, see A synthesis of Solar Cycle Prediction Techniques by Hathaway, Wilson, and Reichmann. Their synthesis works very well, i.e. parameters were adjusted so that past sunspot activity was correlated with current activity. The authors’ model demonstrates a very high degree of precision, and therefore there is a certain level of trust in using the model to predict future solar activity.To some, this regression-type modeling doesn’t seem quite right - as the authors themselves state, some view their modeling technique as only “slightly better than astrology.” This is certainly unfair - if the goal is prediction, then the nature of the model is immaterial - let the best predictor win. At a functional, pragmatic level, it doesn’t matter whether the model for prediction leads to any understanding of the underlying physics. The connection with the Chaos Game couldn’t be more dramatic. Predicting the resulting shapes in the Chaos Game for any shape figure can be handled easily by treating it as a tiling problem, without ever considering the random jumps that are happening on the “microlevel”.In the Chaos Game, however, the rules are known, and unchanging at the microlevel. Therefore it is possible to explain the resulting shapes that occur with a first-principles analysis of the microlevel process. What about the solar dynamo? Can’t fundamental physics be used to create a differential equation model (or model system) that can then be solved for an assumed set of initial conditions? Unfortunately, the state of understanding of the physics of the solar dynamo is still murky - how could it not be, given that turbulent stellar interior?For a thorough review of the status of understanding (or lack thereof) of the solar dyamo as of mid-2005, see the excellent Dynamo Models of the Solar Cycle by Paul Charbonneau. This article contains a very detailed non-linear magnetohydronamic model of the dynamo, and includes a section on the chaotic behavior of the model (warning: not for the squeamish - a lot of nasty Navier-Stokes equations here), but the overall message is that there is still a long way to go before true predictions of the solar cycle will come from this route.

So where do the 11-year cycles come from? Click to enlargeJust when it seemed that understanding was still way off, a new model of solar activity was announced earlier this week by Mausumi Dikpati et.al. (from the National Center for Atmospheric Research in Boulder.) The authors claim an astonishing accuracy rate of 98% in their model predictions! It was this model that predicted the delay in the onset of Solar Cycle 24. Note that the newspaper accounts described in the first paragraph of this post focused on the predictions, not the excitement of the new model. What the news accounts don’t say is that Hathaway, lead author of the statistical prediction paper, disputes the predictions of this new model. According to the empirical model, there will be no delay in the onset of Solar Cycle 24.So who’s right? The beauty of modeling, and the cruelty, is that the next piece of data can either further cement the model’s reputation, or crush it. We won’t have to wait too long to find out.I’m guessing that for the forseable future, solar cycles will be predicted using the statistical method that works so well - and most will worry about avoiding potential problems with satellite communication based on the prediction without worrying at all about understanding the phenomena from a basic physical level.A good website for an account of solar activity and plenty of informative links is NASA’s Solar Physics site, although there is a very disturbing message at the sight claiming that “funding stopped as of october 2005.“An even better site is the Australian Space Weather Agency , which has a large number of constantly updated space weather data, including solar wind speed and x-ray flux. Be sure to visit their Educational area for a very readable overview of solar cycle issues.

Mushroom cloud over Halifax. Click to enlarge. In my previoius post I described some of my thoughts about the teaching of differential equations that model death in some way - via overpopulation, disease, and war - what I call Grim Modeling.Perhaps the grimmest modeling of them all, however, is modeling associated with the design and delivery of atomic weapons - your basic WMD.An essential step in any mathematical modeling is the correlation of the model output with the world, i.e. you need to have some data that supports your model. This is especially true if the model is to be used as a starting point for an extrapolated prediction. By this I mean that t –> infinity, or a parameter is taking on a value much larger (or smaller) than was the case for the situation that produced the data being used to check the model.Could there have been any mathematical exercise more fraught with danger than the modeling done by J.R. Oppenheimer and his team of scientists on the Manhattan project? In a very short time (the summer of ‘42 ), a group of physicsts, chemists, and engineers managed to develop the theory of nuclear reactions in an atomic bomb, the engineering required to design and construct a deliverable weapon, and model the effects that the atomic blast would have on a city and its population.Model the effects of an atomic blast? What type of model was this, and, more important, what data could possibly be used to validate the model as one that could be trusted? This was not an idle concern - some on the Manhattan Project believed that an atomic bomb blast would start a chain reaction that would spread through the earth’s atmosphere - in effect blowing up the earth.Oppenheimer’s search for blast data took him to a most unusual location: Halifax, Nova Scotia. In a terrifying incident that is still very little known (at least outside of Canada), a French munitions ship exploded in Halifax harbour towards the end of World War 1. The ship was loaded with a highly combustible mixture: 2,300 tons of picric acid (used in manufacturing explosives, particularly artillery shells), 200 tons of TNT, 10 tons of gun cotton (which has a history of spontaneous explosion) and 35 tons of benzene.The blast is claimed to be the largest man-made explosion before the atomic bomb. The ship itself was reported to be “vaporized” - nothing remained in the harbor, but pieces of the ship were found embedded in the walls of buildings through the city. The anchor was found 2 miles from the blast site. Up to 2000 people died, many of them children in schools that ringed the harbor area. The explosion produced what is believed to be the first mushroom-cloud (seen in the above image) . The tales of shock, misery, and grief from that time are eerily parallel to ⁹⁄₁₁.

The story goes that Oppenheimer and his group of theoreticians and engineers met at Berkely during the summer 1942 to carefully go over all aspects of the potential atomic bomb known at that point. In addition to the physics and chemistry and engineering, they needed to predict the yield of the weapon. Oppenheimer and his crew used the reports of the Halifax blast damage to model the destruction that would be caused by the atomic bomb. Scarily, they had to multiply the Halifax effects by many orders of magnitude.More scary, the estimates were very close to the mark.No wonder that Openheimer’s thoughts on witnessing the first succesful testing of an atomic bomb comes from the Bhagavad Vita: “I am become death, the destroyer of worlds. “You can view a film clip of Oppenheimer describing the blast and his quote at the Atomic Archives.For more on the Halifax Explosion, see the recent book by Laura Mac Donald titled “Curse of the Narrows”. The book has received very good reviews, with claims that it is the “definitive account” of the disaster.Finally, as a finalist in the Totally Inappropriate Team Name category, there is a fantasy football team named The Halifax Explosion in the Bay Area Custom Football League.

Battle of TrafalgarI often wonder how appropriate it is in an Intro. Diff Eq. course to discuss the darker side of modeling - the business of modeling death and destruction via war, terrorist activity, or plain, old-fashioned man-made disaster. (I have no qualms about this in our Chaos and Fractals course - after all, one of the goals of the course is to consider the ramifications of the pervasiveness of mathematical modeling in all aspects of society, and especially in areas where religion, philosophy, and politics intersect.)But back to Differential Equations - basically a mathematics course where students get their first real concentrated exposure to applied mathematics and modeling via basic calculus.It seems OK to discuss population models such as the Logistic equation, where growth rates go negative for population values exceeding the carrying capacity of the ecological niche. The logistic equations is, after all, a sterile shorthand for what is an accepted “law of the jungle,” at least for animals: if there ain’t enough food, you die.And there doesn’t seem to be any queasiness among students when a Lotka-Volterra model is used for a Predator-Prey system. This again may be explained because it is a fact of life on the other side of the human/animal divide - eat and be eaten, in an endless cycle of life and death. Or maybe it’s because the use of cute function names - R(t) and F(t) for rabbits and foxes - provides a welcome relief from reality due to the abstractness of the notation: we don’t feel the rabbits’ pain, or wince at the growth of the Rabbit-Fox interaction term when dR/dt<0. Epedemiology is a fertile source of models for Diff. Eq., but even here what is being modeled may be especially stark. (See Deterministic Modeling Of Infectious Diseases:Theory And Methods, a thorough review of disease modeling by Helen Trottier and P. Philippe of the Univ. of Montreal). In this case, an ambiguous use of English can go a long way towards not having to think about the grim reality of what the models often predict.Consider the simple SEIR model for time-evolution of infection disease, where the acronym for the dependent functions comes from

• S: susceptible
• E: Exposed
• I: infectious
• R: removed or recovered

HIV Color Game
I recently came across some interesting variations and applications of the Chaos Game.

One version is located at the Boston U web site and is credited to Johanna Voolich and Robert L. Devaney. Their version is very unique twist on the Game - instead of the randomness that forms the basis of the Game, they have replaced the randomness with a strictly deterministic game. In their version, the player tries to get a random starting point into a specific target area of the Sierpinski triangle by choosing a series of half-way jumps to specific vertices. The "winner" of the gme is the one who reaches the target area in the fewest number of jumps. (Click here to play the game on-line.)

An even more fascinating Chaos Game comes from Dan Ashlock at Iowa State University, where he uses a 4-cornered Chaos Game to display the sequence of bases in DNA molecules. (The image at the top of this post is from a segment of HIV DNA.) In this Chaotic Game of Life, points are drawn half-way to vertices depending on the next base in the DNA strand. The pictures look remarkably similar to playing the Chaos game randomly! See Ashlock’s website for a set of images from different organisms, as well as some Markov chain models that help explain and interpret the images.

I just finished two separate presentations of the Chaos Game exercise that I typically use to start the Honors class to students at Arcdia University. Most of the students were education majors - and most of them were elementary ed majors. I am always pleased at the reactions to the game - especially from students who believe, or claim to believe, that they are not good at mathematics. The Chaos Game is such a rich exercise, especially given the approach of Understanding vs. Prediction, that most get caught up in the inadequacies of prediction, and the frustration of not being able to get a real comprehension of how the patterns (and colors) arise.

But the best part of the exercise, for me, is the ability to run it in almost any setting - from a 3rd grade classroom to a university mathematics department. Each group will get something different from the exercise. Personally, I prefer the response of the 3rd-graders, who , unlike mathematics faculty, don’t try to understand what is happening at a deep mathematical level. Instead, they react purely to the shapes and colors, and their inquisitiveness makes for a wonderful teaching moment.

Education majors, and especially elementary education majors, are also a terrific audience for the Game as we play it. I believe that most of them could see how the exercise, or something similar, can be transported to their own future classrooms!

One of the best "teaching moments" I experience teaching the Chaos and Fractals course is when students get enmeshed in BIG issues and start linking the concepts presented in their readings and discussed in seminar to the various uses - good and bad, utilitarian and political, of modeling in the world.

In the past two versions of the course, the science and politics of climate change has been one such issue. This is mainly because both courses were done while the Kyoto agreements were very much in the news - I taught two versions of the course that straddled the change in presidency changed from Clinton to Bush, with the concomitant refusal to sign Kyoto and strong attempts to dismiss the science of global warming predictions and causes. The class discussions that term indicated to me that students were starting to grasp the full impact of how we come to learn about scientific issues from the media and, in turn, how the media’s presentation is ultimately channeled by prevailing political ideology and efficient spin doctors.

Until recently, I didn’t have a really good context and source for helping out these class discussions, other than my own bristling at what I saw as clear anti-science stance taken by the current administration. I have since been reading about the work of Matthew Nisbit, in the Communication Department of Ohio State. Nisbit is a communication theorist who specializes in what he terms Framing Science (also the name of his blog) In his words:

At FRAMING SCIENCE we track how political strategists, scientists, and the news media selectively define science in ways that shape policy decisions, public opinion, and political culture. We apply "framing analysis" to understand the social meanings behind technical controversies (and sometimes we will look at other areas of politics.) Frame analysis is an incredibly useful invention of the social sciences, diffusing across a number of academic disciplines. Frames are used on an everyday basis by political operatives, journalists, and average citizens (though they may not realize it.)

I could have really used this blog as a resource this past semester, when the Evolution vs. Intelligent Design Debate was clearly being "framed" from all sides of the political spectrum.

So add the Framing Science blog to the must-subscribe-to list (especially in conjunction with the science/politics blog The Intersection, described in my previous post.). At the very least, it will give me, and my future students, a special resource to "frame" our own discussions and understanding of whatever the scientific/policy debate happens to be during that term.

I’ve come across a number of very interesting blogs that are non-linear mixtures of science/culture/politics/everything else. Some are solo efforts, while others are collaborations among scientists - something I hope that FractaLog can become…

In the meantime, there are some fascinating, and provocative posts. Please visit … RealClimate

"RealClimate is a commentary site on climate science by working climate scientists for the interested public and journalists. We aim to provide a quick response to developing stories and provide the context sometimes missing in mainstream commentary. The discussion here is restricted to scientific topics and will not get involved in any political or economic implications of the science."

ScienceBlogs This is a collection of blogs hosted by Seed Media Group (they publish Seed magazine, a magazine devoted to science and culture). From the ScienceBlog site: "ScienceBlogs is the web’s largest conversation about science. It features blogs from a wide array of scientific disciplines, with new voices coming on board regularly. It is a global, digital science salon."

As of this posting, there are 15 blogs hosted at ScienceBlogs. My favorites:

• Cognitive Daily (with timely reports on advances in cognition theory and practice)
• The Intersection (On the intersection of science and politics
• Uncertain Principles ("Ramblings about life as a physicist on the tenure track at a small liberal arts college.")
• Adventures in Ethics and Science
• Pharyngula ("Evolution, development, and random biological ejaculations from a godless liberal.")

Cosmic Variance This blog is maintined by a group of physicists and astrophysicists… from different universities. Although dedicated to science, the bloggers do post regularly on " arts, politics, culture, technology, academia." Note that there is also a good list of Physics blogs.

Stretching Field: lines of large past (red) and future (blue) stretching. Click to enlarge
Jerry Gollub of Haverford College is understandably quite famous for his pioneering work in measuring the onset of turbulence. With his conceptually clean (yet technically difficult because of their precision) experiments, he and his colleagues and students have produced a wide range of experimental and theoretical results that demonstrate the role of chaotic dynamics in fluid dynamics.

Gollub is at it again, this time with colleague Paulo Arratia of U. Penn. Gollub and Arratia designed a clever experiment in which they were able to observe the mixing of two fluids in a regime known as "chaotic advection," which is distinctly different from turbulence. (See the review article Mixing, Chaotic Advection, and Turbulence by J.M. Ottino for a good description of these different fluid regimes.

As described in the Feb., 2006 issue of Physics Today (and soon to be published in Phys. Rev. Lett.), Gollub and Arratia were able to measure the stretching field of their fluid. This field is the "local distortion of an infinitesimally small fluid element." This field, in turn, can be used to calculate the Lyapunov exponent for the fluid under different mixing conditions. (The Lyapunov exponent is a well-established measure of the tendency for the phase trajectories of chaotic systems to move apart.) Remarkably, Gollub and Arratia found that they could model the amount of chemical product formed from their mixing reactants as a function of Lyapunov exponent only for a large range of mixing conditions. This result is important because it demonstrates yet again one of the hallmarks of chaotic systems - universality, which is Feigenbaum’s contribution to chaos theory (and which the Gollub/Swinney rotating cylinder experiments helped establish as experimental fact.)

To view more ongoing Gollub projects (as well as interesting applets showing chaotic mixing), visit the Nonlinear Physics and Fluid Dynamics Lab of Haverford College.

I’ve really been happy by student response to blogging in the Chaos and Fractals course (although I didn’t start up the blog until late in the semester, with a resulting small number of student posts.)I’ve been thinking a lot about how to improve on this blog, and also wondering about other blogging opportunities for different classes - my own, and for other disciplines.There’s a wonderful blog devoted to Edublogs - blogs for education. Titled The Edublog Awards: Awards for scholarly and education focused bloggers the blog (managed by James Farmer ) has given out awards to the top edublogs - as voted on by educators.Check out the 2005 Winners. The categories include

• Most innovative edublogging project, service or programme
• Best newcomer
• Most influential post, resource or presentation
• Best designed/most beautiful edublog
• Best library/librarian blog
• Best teacher blog
• Best audio and/or visual blog
• Best example/ case study of use of weblogs within teaching and learning
• Best group blog
• Best individual blog

While you’re at it, check out Incorporated Subversion, James Farmer’s blog. Farmer is a designer, educator, writer and consultant living and working in Melbourne, Austra …At his blog you’ll see several of his initiatives, including free blogs for university students (uniblogs), educational blogs (edublogs), and social software and open source for education.

(Note: I got the idea for this post after reading a nice summary of the Edublog awards by Joyce Kasman Valenza in the Jan. 22, 2006 Phila. Inquirer)