Ancestors

Written by John Carlos Baez on 2025-01-08 at 17:41

When enough electrons get ripped off the molecules of a gas, it can become so electrically conductive that long-range electric and magnetic fields dominate its behavior. Then you've got PLASMA.

Plasma rules the world of astrophysics. It spans an enormous range of densities and temperatures, from interstellar space to the Sun's core.

For some reason I never seriously studied plasma until a few weeks ago, when the Parker Solar Probe penetrated the boundary separating the solar wind from the Sun's upper atmosphere - its corona.

Trying to understand this, I started reading about the equations of 'magnetohydrodynamics'. These are a combination of the equations for electromagnetism and the equations describing fluid flow. Not all plasmas are well described by the equations of magnetohydrodynamics - they're approximate - but many are. And these equations describe a bunch of weird things that plasmas do!

First of all, in these equations the magnetic field is generally more important than the electric field - as the name implies.

Second, when the electrical conductivity of the plasma is very high, the magnetic field tends to get 'frozen in' to the plasma. In other words, you can visualize the magnetic field as a bunch of 'field lines' that move along with the flow of the plasma.

But third, these magnetic field lines have pressure: parallel field lines tend to push each other away. And they have tension: curved field lines tend to straighten out!

And as the field lines do these things, they push the plasma around.

The math of this is pretty fascinating. The equations are terribly hard to solve, but beautiful to contemplate.

(1/n)

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Written by John Carlos Baez on 2025-01-08 at 17:50

Here's my little initial foray into the wonderful world of magnetohydrodynamics:

https://johncarlosbaez.wordpress.com/2025/01/01/magnetohydrodynamics/

I show how to derive the basic equations, and I use them to explain magnetic pressure and magnetic tension. All this stuff is standard. I just never learned it in school! I was too enamored with the charms of pure mathematics and 'fundamental' physics.

If you're scared to look at my article, I'll still show you the key equations. (Do my next posts need a content warning for the math averse?)

The animated gif here is from

• Philip Mocz, Create your own constrained transport magnetohydrodynamics simulation (with Python), https://levelup.gitconnected.com/create-your-own-constrained-transport-magnetohydrodynamics-simulation-with-python-276f787f537d

(2/n)

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Toot

Written by John Carlos Baez on 2025-01-08 at 18:06

Here are the equations of magnetohydrodynamics, or MHD for short!

The fields here are

• the velocity of our plasma, 𝐯

• the density field of our plasma, ρ

• the electric current, 𝐉

• the electric field, 𝐄

• the magnetic field, 𝐁

The quantities in boldface are vector fields while the density ρ is a scalar field. I'm assuming the plasma's pressure is some function 𝑓 of its density, so that's why you see 𝑓(ρ) in the equations. There are also three constants:

• the magnetic permeability of the vacuum, μ₀

• the electrical conductivity of the plasma, η

• the viscosity of the plasma, μ

But my blog article explains step by step how we get these equations, and a few basic things we can do with them. That's the fun part.

I do not explain how the magnetic field can get 'frozen in' to the plasma. Maybe I'll do that some other time. I'm not completely happy with the usual story about that, so I'd like to expand on it a bit.

(3/n, n = 3)

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Descendants

Written by Helena on 2025-01-08 at 19:00

@johncarlosbaez enjoyed the article! Whenever I read about magnetohydrodynamics it always feels like the universe being cheeky and saying "atmospheres (which you people have spent over a century trying to model) are fantastic, but not complicated enough"

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