Magneto-what?

Today’s post is a quick one, since quantum mechanics and abstract algebra (in particular) and classes (in general) have kind of taken over my life. The subject is something that I got interested in about a year ago, in an advanced applied math class of all things: Magnetohydrodynamics, or MHD for short.

The basic idea behind MHD is relatively straightforward. One of the basic principles of electromagnetism is that electric and magnetic fields are intimately linked in a manner that is encapsulated quite elegantly in Maxwell’s equations; for example, a changing magnetic field can give rise to an electric field and induce a current in a conductor. One of the basic principles of fluid dynamics is that, well, fluids move in an interesting way, and the starting point for understanding this is the Navier-Stokes equations; for example, turbulence manifests itself in myriad ways, from smoke rising to ocean currents to the atmospheres of stars, and is still not very well understood.

Very well then: what happens when you have a charge (e.g. ionized) fluid whose dynamics are describable by the Navier-Stokes equations in the presence of a time-varying magnetic field? For example, the field affects the overall macroscopic motion of the fluid, which in turn gives rise to further electromagnetic effects since it is charged. How does one understand this kind of system? This question is the heart of MHD, a subject initiated by Hannes Alfven – for which he received the Nobel Prize in 1970 – and one that has been the subject of research in many, many fields, including applied mathematics, plasma physics, astrophysics, and geophysics.

The sun is an excellent example of a rich MHD system (the image above is from here, by the way): everything from sunspots to the solar wind to the Parker spiral involves magnetohydrodynamical effects. (That last one is particularly fascinating: did you know that the spiral shape of the Sun’s magnetic field “as it extends through the solar system… is similar to the pattern produced by a spinning lawn sprinkler, for similar reasons”? I didn’t before I came across MHD.) Another system that seems to be under quite intensive study at the moment is the magnetorotational insability (MRI), which is apparently a means of explaining anomalous viscosity in accretion discs.

Anyway, the mathematical details (which I had to work out for said applied math class and were surprisingly fun) get slightly complicated, but the key principles of ‘ideal’ MHD are that:

• the charged fluids are continuous, so one can ignore mean free path effects
• the charged fluids are perfect conductors and flow without drag
• thermal effects don’t matter
• the speeds under consideration are small enough to ignore relativistic effects

Now clearly these assumptions are whopping ones, and do indeed break down in many circumstances – but that’s ok. A good deal of hard work by a number of people has led to immense progress in understanding ‘non-ideal’ MHD, but it’s surprising how far you can get with these simplifying assumptions. One could, for example, start with the Navier-Stokes continuity (basically a way of saying that mass is conserved) and force (a way of saying that momentum is conserved) equations, plugging in expressions for electric and magnetic fields, and throwing in the energy equation (the first law of thermodynamics, which is a way of saying that energy is conserved) and the induction equation (one of Maxwell’s equations, which specifies how the magnetic field changes with time). But wait, we’re not done: to ‘close’ this system of equations, one can throw in two more equations: another of Maxwell’s equations (the divergence-free condition for the magnetic field to make sure you have no magnetic monopoles floating around) and the ideal gas equation of state. Turn the crank, and out pops a new field of study – MHD. So there you have it.

As an aside, I want to briefly point out that one of the reasons why this and other things of a fluid dynamics nature are of interest to applied mathematicians is because most of the time, one can’t solve these equations exactly and the analysis can be very tough (in math-speak, these equations constitute a nonlinear hyperbolic system, which means that finite-difference schemes tend to give solutions that blow up, among other things). Mathematicians can be quite good at coming up with ways of coping with this – for example, coming with means of simulating them numerically (such as Godunov’s scheme).

Atomic Spectroscopy

I’m doing an ‘experiments in modern physics’ lab this semester. The structure of the course is simple: we do 4 labs chosen from a tantalizing list of many, complete with a report and an APS-style talk. I decided to do my first experiment on atomic spectroscopy: measuring energy level splittings using optical resonators, and I figured I’d write a post on it (well, paraphrase my report) because it may be interesting to a non-scientific audience. So, this post only has a few equations and is mainly an overview of the basic concepts involved.

(For those interested in the details, here’s the executive summary: the experiment was basically to use Fabry-Perot and Michelson interferometers to characterize the spectral lines of atomic sodium and hydrogen/deuterium, and measure the spectral splitting of the sodium 3p_3/2→3s_1/2 / 3p_1/2→3s_1/2 D line and the hydrogen/deuterium 3s→3p H-α transition due to spin-orbit coupling and isotope mass differences, respectively. The H/D splitting measurement also allows one to deduce the proton/deuteron mass ratio by playing around with the numbers a bit. Note that the picture on the left is a photo of slightly split sodium fringes from a Fabry-Perot interferometer, from [1] – click Read on… and scroll to the bottom for the references.)

In this post I’ll briefly review the historical context of this experiment, go over the basic physics responsible for the spectral splittings and the physics behind the experimental tools, and outline the experiment itself.

Curious Math Word Of The Day: Germ

From Using Algebraic Geometry, by D. A. Cox, J. Little, and D. O’Shea:

Two analytic functions, each defined on some neighborhood of the origin, are equivalent if there is some (smaller) neighborhood of the origin on which they are equal. An equivalence class of analytic functions with respect to this relation is called a germ of an analytic function (at the origin).

This is a pretty good word to use for this purpose, given the meaning, but a peculiar choice of language nonetheless.

(Walt from Ars Mathematica points out that the word comes from wheat, which makes sense given its meaning. Other examples of agriculture-based math terminology that I can think of are the “kernel” of a map, or the “fibre” of a map.)