Transitory Turbulence

Turbulence is often called the last unsolved problem of classical mechanics. It is a ubiquitous phenomenon that anyone who has ever flown on a airplane knows about, but there have been exactly two major breakthroughs in turbulence theory in the past 70 years. Both of these breakthroughs came in the early 1940's and both were made by a Russian mathematician and physicist named Andrey Kolmogorov. However, there has been a lot of work on the problem, especially since the development of the theory of dynamical chaos in the 1990's. One of the big questions, in the language of dynamical chaos, is whether turbulent states are attractors or repellers. Put another way, if you get a system into a turbulent state, will it remain turbulent indefinitely or will it eventually decay into a laminar state? There are, of course, three possible answers to this question: yes, no, or maybe.

We can throw out the yes answer by common counter examples. If you pour water into a glass the flow is normally turbulent, however it quickly becomes laminar if you stop pouring. The real debate then, is whether turbulent systems always decay into laminar systems or if there are some special cases where turbulence is a chaotic attractor.

Before we go further, I need to introduce you to a parameter known as the Reynolds number. Fluid dynamics is full of dimensionless parameters and in turbulence the Reynolds number (abbreviated Re) is the most important one. Essentially, the bigger the Reynolds number the more turbulent the system is. More specifically, the Reynolds number is the ratio of inertial forces to viscous forces. In laminar flows, small chaotic motions are damped by viscous forces (low Re), while in turbulent flows viscosity is insufficient to damp out the small chaotic motions (high Re).

The maybe answer comes from the argument that above some Reynolds number, turbulence is self-sustaining. However, a new paper in Physical Review Letters by Bjorn Hof, Alberto de Lozar, Dirk Jan Kuik, and Jerry Westerweel (and nicely summarized by the APS at http://physics.aps.org/) claims that the correct answer is that there is no such thing as a turbulent attractor. They conduct a number of experiments that show that the decay time from turbulence to laminar flow goes as a super-exponential function of the Reynolds number ( tau = k*exp[exp[Re]] ). This trend fits very nicely over 6 orders of magnitude in decay times from ~5 seconds to ~150 days as seen in the figure to the right. This gives strong evidence that turbulence always decays into laminar flow, it just takes a while in some cases. In fact, for a system with a Reynolds number of 2200, the decay time is longer than the age of the universe.

By comparison, in the solar convection zone the Reynolds number is something like 10^10. That's a decay time of 10^[10^(10^9)] seconds. As they say in Brazil, don't wait standing.

While the debate may not be concluded quite yet, it appears that turbulence is really a temporary phenomena - if by temporary one means it commonly lasts orders of magnitude longer than the age of the universe. More meaningfully, it means that from a theoretical standpoint, turbulence is a chaotic repeller rather than a chaotic attractor. Perhaps most importantly, this means that we are still making progress on the last major puzzle of classical mechanics.

The Solar Dynamo

This is part two of my series of posts on the exciting field of solar physics. For part one, click here.

**************************************

Despite centuries of study, we fundamentally don't understand how the sun generates its magnetic field and magnetic cycles of activity, but we do have some ideas. Here's what we know about the way the sun (and by extension other stars) produces and varies its magnetic field.

The source of the sun's magnetism is some kind of dynamo process. Dynamos occur when a highly conducting material shears against itself in the presence of a magnetic field. There are several requirements for this to occur. First, the material must be a very good conductor, such as the ionized plasma in the solar interior or the liquid iron in the earth's core. A good conductor means that charge carrying particles are essentially free to move through the material. When a free, charged particle encounters a magnetic field line, it will begin to move along that field line in a helix pattern. The magnetic field forces the fluid to move along it, while the fluid circling the field line creates a current that reinforces the magnetic field. This is referred to as the frozen in condition because the fluid can only flow along magnetic field lines and the magnetic field lines are continually regenerated by the motion of the fluid. In non-superconducting materials, like the solar interior, this is an imperfect process and it is not strictly true, howeverthe frozen in condition is still a good approximation.

In the solar convection zone, magnetic fields exist in the middle of extremely turbulent convection. When the convective motions cause motion of the fluid along a magnetic field line, they stretch the field line, much like taffy stretches when you pull it. This stretching puts energy into the magnetic field, causing it to grow stronger. The solar interior is therefore one gigantic, fusion-powered taffy pull which constantly regenerates the sun's magnetic field.

I should also mention that dynamo processes are inherently non-linear. The "taffy pull" effect requires advection, which mathematically comes in the form of the gradient of the velocity squared. That term (and a couple other non-linearities) cause me to periodically wake up at night in a cold sweat. Because of the non-linear properties, dynamos are both generally chaotic and almost impossible to work with analytically (although some brave people like Matthias Rempel at the National Center for Atmospheric Research try anyway). This means that almost all theoretical work must be done numerically.

So a dynamo seems like a nice theoretical construct for the source of the sun's magnetic field, but is that actually what is going on? And what about those cycles of magnetic activity? Can a dynamo explain that butterfly diagram from the last post? Tune in to my next post and we'll talk about how we investigate what is actually happening inside the sun.

Why Study the Sun?

When Joe posted about changing his blogging style, it got me thinking about my past blog entries. As I pondered, I realized I really never explained what it is that I actually do. So my next several posts will focus on explaining what solar physics is, why it is important, and what the future may hold for my field of research. Enjoy!

****************************

Perhaps the place to start is with the one of the most famous diagrams in astrophysics - the butterfly diagram (this version courtesy of NASA).As you can see, there is a distinct, repeated pattern in the number and latitude of sun spots. Sun spots always come in twos and are caused by strong magnetic fields poking up through the solar surface. These strong, coherent magnetic fields stifle convection in the sun spot, causing in to cool below the temperature of the rest of the solar surface and thus look darker. What the butterfly diagram shows us is that the sun experiences 11 year cycles of magnetic activity.

Why does this matter? For one, these cycles indicate that the highly turbulent convection present in the sun produces ordered, regular phenomena on global scales. This is a classic example of the famous quote by the Nobel laureate condensed matter theorist Phillip Anderson "more is different" (incidentally, Anderson also initially proposed the Higgs mechanism for particle mass). This is something like saying that you can get a pot of boiling water to produce repeated patterns where bubbles start at the edge of the pot, move inward at a predictable rate, and then do it over again. The problem of solar magnetic cycles is a fundamental problem of non-linear dynamics - small, chaotic actions lead to large, organized structures.

The second, more practical reason to care what the sun's magnetic field is doing is summed up quite nicely by the picture below.You may not realize it, but the sun is trying to kill us all. Luckily, we have 92 million miles of separation, the earth's magnetic field, and an atmosphere to protect us, but as people move out of the Earth's protective cradle, there is a increased need to understand why the sun goes through these cycles so that we can better predict when and where these massive blasts of highly ionized plasma will strike.


So why do I study the most studied celestial object in human history? Because despite all of the intelligence that has been thrown at the sun over the ages, we still don't understand fundamentally why it does what it does. We don't understand solar magnetic activity for the same reasons we don't understand things like structure formation in the universe, weather forecasting, or the molecular structure of solids - they are all fundamentally non-linear problems.

And aside from all the deep reasons to study the sun, it is also trying to kill us...