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The topic of my physics undergraduate research thesis. It is something that, to the best of my knowledge (and my professor's), no one else had studied as deeply as I had. So, unless someone else has done it, I'm the world's foremost expert on something almost entirely useless.

I would just put my entire thesis up here and let people sift though fifteen pages of physics mumbo-jumbo, but few would understand, since it isn't written for laypeople, and I wouldn't be able to include the many diagrams and graphs, so it would lose a lot there. Also, I doubt anyone would read through the whole thing, since it's fairly dry.

So, I will attempt to summarize as briefly and coherently as possible, while capturing the key facts.

A vortex is what most people would commonly call a whirlpool if it occurred in water, although they can and do occur in any fluid. The vorticity of a fluid at a certain point is defined as the curl of the velocity field at that point. That is, you find out how the fluid is moving at every point, and apply a certain vector operator called curl to that to find this quantity that basically tells you how much the fluid is spinning.

Vorticity is a real pain to work with (actually, just about everything in fluid dynamics is a pain to work with), so a common approximation is to imagine that all the vorticity for a given vortex is concentrated into a single line; so the vorticity is zero everywhere, except along a single one-dimensional line, where it is infinite. You can still control how strong the vortex is by defining your infinity by a certain kind of limit of a series of functions, known as a Dirac delta function. The constant out front of said function is the vortex strength, usually written as a capital gamma.

Now, say you have a bunch of line vortices, all running parallel to one another. Any cross-section of the system perpendicular to the axis of these vortices will be the same as any other, so we can reduce the system to two-dimensions, and call the line vortices point vortices instead.

A point vortex exerts a velocity influence not only on the surrounding fluid, but on other point vortices as well. So a system of point vortices will move around under their mutual influences in a manner somewhat analogous to a system of Newtonianly gravitating bodies. The crucial difference here is that under Newtonian gravity, bodies exert an acceleration on one another, and that acceleration is parallel to the line connecting them. Vortices exert on one another a velocity influence, rather than an acceleration, and that influence is perpendicular to the line connecting them. Therefore, the differential equations describing the system are first order, not second.

This difference creates a significant differences in the chaos properties of the system. For Newtonian gravity, chaos arises when you have three or more bodies. For point vortices, three of them will follow closed paths and therefore have analytic solutions to the equations of motion. This system has been thoroughly studied. Chaos arises when you have four, hence my choice to deal with exactly four for my thesis, as the lower limit for chaos.

So, what did I do? Working with actual point vortices is nearly impossible, since point vortices are meant to be an approximation to a physical system, not an exact representation. Therefore, they don't exist, except in superfluids, and studying four point vortices in something like liquid helium was a little beyond the capabilities of a fourth-year physics undergrad.

Therefore, I did what physicists usually do in this kind of situation. I made a computer simulation.

The simulation program, written in Java (because that happens to be the language that I knew), uses fourth-order Runge-Kutta numerical integration to approximately solve the coupled first-order linear partial differential equations and spits the results out as a string of raw numbers in a format that can easily be imported into Microsoft Excel. Using Excel, I then made charts of the vortex trajectories, tested for chaos (using the idea of Lyapunov exponents), and then tried to look for patterns in the chaos. The charts are actually quite beautiful, resembling the results of a Spiradoodle gone mad.

My findings:

  • The system is indeed chaotic, as predicted. That is, no analytic solutions exist to the equations of motion. A small disturbance in initial configuration leads to long-term qualitative differences in behaviour.
  • There are a very small number of highly symmetric configurations that are stable and possess analytic solutions due to their symmetry. Any deviation from perfect symmetry leads to exponential growth of the deviation, however, so there is no conflict with the definition of chaos.
  • For most configurations, each vortex is confined to a finite region in the shape of a ring or disc. Each vortex will be ergotic within its region. That is, given an infinite amount of time, it will pass infinitely close to every point in that region an infinite number of times. Put another way, the trajectories never close. A vortex's motion never repeats itself for four or more vortices.
  • This is the most important and impressive bit, that I discovered in a "eureka!" moment about one week before the thesis was due. If there is some subset of the vortices, for whom the total strength is exactly zero (vortices can have positive or negative strength, depending on whether they go clockwise or counter-clockwise, determined by the right hand rule), then those vortices will always eventually form a group together and meander off towards infinity together, leaving the odd men out behind. This is because it is impossible to define a center of vorticity for those vortices, and so they are missing one of the crucial Hamiltonian constants of motion necessary to restrict them to a specific region. In this case, the system is non-ergodic, and this is the only exception to the preceeding rule. Note that sometimes the vortices will head off together almost immediately, but sometimes it takes a long time. Included in my original thesis paper is a chart of the trajectory of one vortex in a group that took approximately 250,000 seconds to start their journey into the infinite beyond.

None of these discoveries is particularly useful, although I discovered a simple way to estimate an upper limit on the size of the region of ergodicity for the non-exceptional cases (the ones that don't fall into the category mentioned in my last discovery), which means that if you are dealing with four vortices, you can check if you are "sufficiently far" out of the region they are free to wander, and if so, treat them as just a single vortex with strength equal to the sum of their individual strengths. Not a situation you encounter every day, but still something that theoretically has potential use.

If I had decided to continue on to graduate school in physics, I probably would have elaborated on my thesis to try to come up with something with practical use. As it is, I'm teaching English in South Korea, and will probably never return to physics. Therefore, The Four Vortex Problem will probably just continue to gather dust on my hard drive, in the archives of Queen's University (if they even keep undergrade theses... I don't know if they do), and now, in the nodegel of E2. My profs were impressed, though, and hopefully some of you will find it interesting, too.

I know I promised to append the Java code for my simulation, but upon trying to do so, I realized the extreme pain in the ass of trying to make a few hundred lines of code look nice on E2, what with getting things indented properly, making the square brackets show up, etc. An easier solution: If you want the code, email me at "alex at omniheurist dot net."

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