### Early Stormy Thoughts

Turbulence is the chaotic motion of air and water. Turbulence characterizes hurricanes and tornadoes and ocean waves breaking onto shore; turbulence is the smoke that rises from a cigarette or the movement of air during airplane flight that causes the plane to jounce and me become alarmed. Turbulence is a “wild erratic dance of fluids.” Drip by drip the water drops from my faucet and each drop lands exactly in the same location., but the more I turn the handle the more turbulent the flow becomes. Suddenly drops appear somewhat randomly on the sides of the sink: a drop here and a drop there. Why? How? How there? Why now? I think we have the capacity to predict the movement of water in a lazily flowing river, but the challenge of the movement of the rapids assumes uncertainty. Somehow, in a situation, depending on speed, pressure, velocity, viscosity, etc., the particles interact in ways that are ultimately unpredictable. Where is this storm heading: we remain essentially unsure. There would be great benefit to human life (and safety) if we were able to to predict the movement of such flow for such phenomena as hurricanes, etc. But, in fact, we cannot do so with any high degree of certainty. In fact, in my experience the weather people are fairly consistently inaccurate: they predict a heavy snowfall and no snow falls; they warn that a hurricane should strike land at a certain hour and alas, with good fortune the storm heads out to sea. The ten-day forecast is usually good for at least several hours! We imagine that everything about our natural world will follow predictable patterns and that formulas can be written to enable accurate prediction. But as yet the problem of turbulence has not yet been solved.

The Navier-Stokes equation, developed independently by Claude-Louis Navier and George Gabriel Stokes in the early 1800s, measure the motion of fluid—and these formulas, used in a wide variety of instances including weather forecasting, might be used to predict turbulent flow. In the novel

*The Mathematician’s Shiva,*by Stuart Rojstaczer, (2014) the narrator, a PhD topologist and the son of a world famous mathematician who has spend her life attempting to solve the Navier-Stokes formulas, protests, “Understanding a physical phenomenon like turbulence ultimately means predicting its behavior, or at the very least understanding just what can and what cannot be predicted over time. Prediction means quantification of the basic physical processes that drive turbulence. But this is something we cannot currently do with any reasonable degree of sophistication” Navier-Stokes occupies a central place in mathematical culture: a solution to the Navier Stokes would earn its solver the Millennium Prize—a Nobel-like prize in mathematics worth one million dollars. A solution to the Navier-Stokes would show that the Navier Stokes formula works in all cases. So far, however, Navier-Stokes remains unsolved. Turbulence, when a particle becomes chaotic and unstable behaving, cannot be accurately predicted. Richard Feynman has said that turbulence is the last unknown in physics: “ . . given an arbitrary accuracy, no matter how precise, one can find a time long enough that we cannot make predictions valid for that long a time. Now the point is that this length of time is not very large . . . It turns out that in only a very, very tiny time we lose all our information . . . We can no longer predict what is going to happen . . .” Rojstaczer’s narrator, the topologist, says that “Tornados start from small disturbances that don’t mean a thing and almost always dissipate. But somehow one particular random bad event attracts others and all of them grow and attract some nasty stuff.” Tornadoes, hurricanes, and air turbulence cause great tragedy. If we understand turbulence, then we can predict its behavior; thus far all attempts have failed to solve the Navier-Stokes formulas.

Turbulence is the movement of the particles that comprise air and water. Navier-Stokes attempts to predict the movement of these substances that are, of course, comprised of ‘particles.’ Sasha says in the novel, “The movement of such particles depends on so many factors: density, velocity, speed, viscosity, pressure . . . From the standpoint of a mathematician . . . the problem of turbulence is fundamentally one of being certain, to prove, that there exists in the universe a set of descriptors, in this case partial differential equations, that can encompass the behavior of fluids as they move at velocities that cause chaos. . .”. But in fact, the Navier-Stokes formula, which seems to be based on Newton’s second law of motion, has not been solved: we don’t know that it works to describe the motion of fluids “when they flow as lazily as rivers as when they “dance around an airplane wing” (208). Feynman says, “If water falls over a dam, it splashes. If we stand nearby, every now and then a drop will land on our nose. This appears to be completely random . . . The tiniest irregularities are magnified in falling, so that we get complete randomness.” At such moments, I think of the classroom.

In the classroom exists the conditions for the appearance of turbulence. There are particles in movement and interaction both within and between individuals. There is pressure, speed, viscosity, time, etc. How to predict the movement of particles within the classroom may be

*approximated*but never absolutely defined. Every day a teacher walks into her classroom with a diagram (a lesson plan) that (like the Navier-Stokes formula) should predict the direction and movement of the mass of particles, but in fact, like the Navier-Stokes have not as yet been solved.

*If*I apply the formulas (my lesson plans) then every student should move (learn) in a pattern predictable: I should be able to measure that movement with great accuracy. Classes in class room management (themselves not always well managed) teach students how to direct the flow of learning in the classroom. But the existence of turbulence suggests that any small disturbance (any particle or tiny combination of particle events at all) will grow and potentially produce a situation of chaos. Charles Fefferman says concerning Navier-Stokes, “Since we don’t know whether these solutions exist, our understanding is at a very primitive level. Standard methods from PDE appear inadequate to settle the problem. Instead, we probably need some deep, new ideas.” We try to control our lives with rationality, with predictive formulas, “but in fact, the Navier-Stokes formulas have no solutions . . . {and] this difference between our wish for an orderly universe and the reality of the calamity of the natural world makes us deny reality” We imagine a world that doesn’t exist.

I stand in the classroom and the weather is calm. I think. But there are particles in motion and I do not know—I cannot know—which particles will attract others and lead to some rather nasty stuff. My lesson plans deny reality because they offer some false picture of the appearance of order. It is false. Is it possible to make a theoretical model to describe the behavior of a turbulent flow¾in particular, its internal structures? At this time the answer is, no. In the meantime, I stand in the classroom and look out on the horizon for the threat of tornado with only my lesson plan as protection.

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