Van Gogh's Turbulent Period

Starry Starry night,
Vortices swirl around each star,
Eddies within eddies from near and far
With flows that know the darkness in my soul..

OK. Enough Don McClean. This post is about Van Gogh and his uncanny ability to depict reality in an unreal way, raising once again the question: Do certain artists have an ability to capture physical process and /or mathematical truths that can't be mimicked by others? And is there a correlation with this ability and madness?

Where Jackson Pollock's paintings are instantiations of splattered fractals, Van Gogh's paintings have recently been compared to one of the main avatars of chaos theory: turbulence.

Click to read more ...

Sometimes a Great Notion: Turbulence and Ken Kesey

I recently began re-reading Sometimes a Great Notion, Ken Kesey's monumental second novel published in 1964, and one of my favorite books. Even though I read it over 20 years ago, I still remember many of the most famous scenes. More remarkable, perhaps, is the fact that I vividly recall the haunting descriptions of the Pacific northwest so magically captured by Kesey.

When I read about the chaotic aspects of turbulence, especially as described by James Gleick in Chaos: Making a New Science, I am always reminded of Great Notion's opening passage:

Along the western slopes of the Oregon Coastal Range ... come look: the hysterical crashing of tributaries as they merge into the Wakonda Auga River. ... The first little washes flashing like thick rushing winds through sheep sorrel and clover, ghost fern and nettle, sheering, cutting ... forming branches. Then, through bearberry and salmonberry, blueberry and blackberry, the branches crashing into creek, into streams. Finally, in the foothills, through tamarack and sugar pine, shittim bark and silver spruce -- and the green and blue mosaic of Douglas fir -- the actual river falls 500 feet ... and look: opens out upon the fields."

Many others feel the same way as I do about Great Notion. In a 1997 survey, the Seattle Post-Intelligencer published a list of the 12 Essential Northwest Books. Sometimes a Great Notion was #1, after being named the top book by over 1/3rd of the participants.

According to Kesey (who died in 2001), "I think 'Sometimes a Great Notion' is the best thing I'll ever write...Writing it was much different from 'Cuckoo's Nest,' which often seemed like filling in the blanks. 'Notion,' to my mind, is a great piece of work. People sometimes ask me why I don't write something like that again and I reply that I simply can't. I can't keep all that in my head at once anymore. Why, on 'Notion,' I used to work 30 hours at a stretch -- you've got to have youth to do that."

I don't know how the book fares on re-reading, 20 years after my initial reading. Maybe one must also "have youth" to really embrace the novel's sprawling, non-linear, 627 page narrative. I do know, however, that I will always be mesmerized by the "the hysterical crashing of tributaries as they merge into the Wakonda Auga River..."

Chaotic Mixing

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.

Notes on Universality and on Colors

Originally Posted by Tom Plick

Even after Feigenbaum discovered the constant that bears his name, I find it amazing that he could use it to predict the bifurcations of all sorts of different functions - in our experiments, the numbers did not converge quickly, and so there was likely a lot of error involved there.

We talked a lot about the similarities between different types of physical phenomena - for instance, between phase transitions and the onset of turbulence - and we said how the mathematics at the boundary is similar in both cases. This may be, but even if the equations for the two phenomena match up, I still don't see how the wide-reaching concept of universality can pair them. Phase transitions involve changes among the solid, liquid, and gaseous states of matter; turbulence involves a multiplicity of frequencies in an oscillating fluid, resulting in "turbulent or not turbulent". I see no way to parallel these two problems, between three states and two states.

I agree a lot with Goethe's views on color, in particular that perception of color is subjective. I read an article two months ago, talking about linguists' analysis of color words in different languages. In ancient societies, some say, the people had fewer words for colors - the Greeks had only identified a handful. It is interesting to think that perhaps, back then, they actually couldn't see as many colors as we can today. That would be very subjective, no?

Information about the linguistic analysis of color words can be found at the Straight Dope; there is also a nice chart here.