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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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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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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@johncarlosbaez wow magnetohydrodynamics is a topic I quite enjoy. I studied it very briefly in undergrad because it was my undergrad advisor's research specialty. Excited to get into your blog post!
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@JadeMasterMath - wow, I didn't know that. I wish I'd managed to get you doing some combination of physics and category theory, because at that time I was sad that none of my students were doing physics except Blake. (Kenny was studying Markov processes but he didn't really enjoy the differential equation side of the story.)
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@johncarlosbaez Eh at the time categories felt more exciting 👍
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@JadeMasterMath - I'm glad you enjoyed 'em. My friend Robert Kotiuga had a sign on his door saying "Functorial Electomagnetism Lab".
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@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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@helenaisvibing - "let's make the air conduct electricity and turn on a magnetic field!"
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@johncarlosbaez The equations look neat and concise.
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@johncarlosbaez Have you looked at dusty plasma yet? Dust in plasma can arrange in crystal lattice structures. Fun stuff.
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@carapace - I just bumped into a book on dusty plasmas in the library on Tuesday - I did not open it up. Lattice structures??? Wow!
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@johncarlosbaez Yup!
E.g.: "High-speed imaging of dust particles in plasma"
http://dx.doi.org/10.1017/S0022377812000967
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@johncarlosbaez Are the symmetries specific to this simulation, or are there generally symmetries in MHD?
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@grant_h - the equations of MHD have 3d rotational symmetry, but here it's being simulated in 2 dimensions in a square box and it looks like this particular solution just has 180 degree rotation symmetry.
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@johncarlosbaez
I have a couple of questions about your blog post.
Firstly, where can I learn more about magnetic diffusion, or more generally, vector valued diffusion?
I'm very interested in this idea but I can't find a good intuition for the diffusion of a vector valued function in the plane. For example, when the field diffuses out to points where the field was previously zero, is the diffusion somehow the vector average of the inducing non-zero vectors?
I've searched Wikipedia but can't find a nice, basic explanation for how vector-valued diffusion might work.
Secondly, and more trivially, I got confused by the seemingly twice defined quantity in your post shown in the screenshots below. They seem like the same quantity but with two different names (mass current vs momentum density)?
=> View attached media | View attached media
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@TonyVladusich wrote:
"Firstly, where can I learn more about magnetic diffusion, or more generally, vector valued diffusion?"
Since the vector-valued heat equation is just like 3 copies of the usual heat equation, every fact you know about heat equation applies!
The full-fledged magnetic diffusion equation is nonlinear and thus more complicated; you can learn a bit more about it by clicking on "magnetic diffusivity" in my blog, and even more by going to the reference by Ogilvie that I recommend.
"Secondly, and more trivially, I got confused by the seemingly twice defined quantity in your post shown in the screenshots below."
I used two names for it but it's the same thing, ρ𝐯, so I should get rid of one of those names. Thanks!
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@johncarlosbaez
What I find fascinating is that normal matter behaves very similarly at all scales. You can watch the collision of two galaxies or two drops of rain, and they look reasonably alike. But the behavior of plasma in a strong magnetic field looks like nothing we're used to. It looks more like the behavior of living organisms.
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@BartoszMilewski - indeed, the equations of MHD are extremely lively, especially when you throw in gravity and/or nuclear fusion. Maybe I should get one of these gadgets:
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@johncarlosbaez
Your plot of temperature vs density reminds me of something most people find counterintuitive: a practical fusion reactor must be much hotter than the sun.
That's because the sun's fusion reaction rate is very, very slow -- less thermal output than a resting human body!
We can even compute the power output (solar luminosity, human head production) and get the power output per unit mass.
The sun is such a big deal power-wise mostly because there's so much of it!
Source: https://nssdc.gsfc.nasa.gov/planetary/factsheet/sunfact.html
[
\begin{align*}
L_{\mbox{sun}}&= 3.82 \times 10^{26}\ \mbox{watt} \
M_{\mbox{sun}} &= 1.99 \times 10^{30}\ \mbox{kg} \
L_{\mbox{sun}}/M_{\mbox{sun}} &= 3.82/1.99 \times 10^{-4}\ \mbox{watt/kg} \
& = 1.92 \times 10^{-4}\ \mbox{watt/kg}
\end{align*}
]
Source: https://www.researchgate.net/figure/Heat-Production-Rate-in-a-Human-Body_tbl2_271444362
[
\begin{align*}
L_{\mbox{human}} &= 100\ \mbox{watt (60-250 watt)} \
M_{\mbox{human}} &= 50 - 100\ \mbox{kg} \
L_{\mbox{human}}/M_{\mbox{human}} &= 100 / 50-100\ \mbox{watt/kg} \
& = 1-2\ \mbox{watt/kg}
\end{align*}
]
So a human body, at rest, produces around 10^4 times as much power per unit mass as the sun.
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@weekend_editor - that's all very cool. Thanks for typing in all those formulas.
You're reminding me of the joke:
Q: How far away is nuclear fusion as a practical source of power?
A: 90 million miles.
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@weekend_editor @johncarlosbaez A more useful measure is power per cubic meter, which is still pretty anemic at 270 Watts.
So to get 1GW you would need a reactor volume of a bit under 4 million cubic meters, or a cube about 1.6km on a side.
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@tokensane @johncarlosbaez
Yes, though of course nobody would propose using directly the reaction from the sun for fusion power!
That's the Carbon/Nitrogen/Oxygen (CNO) cycle (and numerous variants; turns out the sun's complicated!).
On the one hand, good news: the CNO cycle can happen at much lower temperatures than a tokamak's DD or DT fusion (let alone some wonderful aneutronic beast like protons on boron).
On the other hand, bad news: it's a ring of 6 separate reactions, 2 of which are WEAK interactions tossing off neutrinos.
So that's slow. (There are, I seem to recall, about 5 different variations on this theme at various temperatures and pressures. But always 6 reactions, always 2 of them weak.)
Here's a picture from WIkipedia of one of them (CNO-I).
https://en.wikipedia.org/wiki/CNO_cycle
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@johncarlosbaez this remains my favorite phase plot, as it shows some interesting boundaries
Plasma science: from fundamental research to technological applications, National Academy Press, 1995
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@jaztrophysicist - very nice plot! I'll study it. Someone should create a version with a bit more visual flair (like color).
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@johncarlosbaez Yeah it's quite old now but it's alway good. There's also this one from my colleague Andrea Ciardi in Paris, who is working on the production of experimental high-energy-density plasmas (to which I added the solar wind and Sgr A* accretion flow on it - ICM is intracluster medium, IGM intergalactic medium, ISM interstellar medium, YSO is young stellar object, ICF inertial confinement, MF magnetic confinement)
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@johncarlosbaez but what I find most interesting with the first plot is that it shows the solar interior is actually not so far from being a degenerate, strongly coupled plasma (not in the sense of the strong force, but in the sense of the thermal energy of the plasma being smaller than the coulomb interactions potential energy)
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@jaztrophysicist @johncarlosbaez ok, I'm a bit disapointed because nobody seems to like liquid metals here!
While they are not plasmas they obey the MHD equations. They would lie somewhere near the P and Q of "Quantum Plasmas" or a little bit below.
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@natchouf @jaztrophysicist - I'm sure I would love them if knew more about them! I guess one big difference is that they're nearly incompressible?
I also recently bumped into the terms "electrohydrodynamics" and "ferrohydrodynamics", which hint at regimes I haven't looked at yet.
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@johncarlosbaez @natchouf Yeah ferrofluids are somewhat different, it's yet another branch of hydro. Liquid metals used in MHD experiments are essentially Gallium and Sodium.
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@jaztrophysicist
This is a beautiful and very legible plot! Which tells me that it was not done by an astrophysicist 🙃
@johncarlosbaez
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@j_bertolotti @johncarlosbaez Plasma physicists are qualified physicists with very good average maths credentials, yes. They had to study hard to figure out how to make the H-bomb.
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@jaztrophysicist @johncarlosbaez oh, I remember that from when I was silly and chose fusion reaction as math spe project.
When I learned I was definitely not good enough in mathematics to follow that path.
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