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The Euler equations produce singularities (in finite time) under certain conditions see here

When rings of fluid approach each other,  in certain simplified scenarios,  the Euler equations that describe their motion break down. 

Nevertheless,  even in more realistic (non-idealized) models, although singularities will not appear,  regions of extremely high pressure (density, vorticity) will emerge.

Question.  Could this work be extended for the MHD equations governing the behavior of plasmas? In simplistic terms, could two (high density) rings of plasma colliding lead to the emergence of regions of high density (temperature, pressure) plasma, that could facilitate fusion in a tokamak (or stellarator),  for example? See comments for (heuristic) details.

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    $\begingroup$ Could MHD equations lead to singularities? Possibly, though I imagine the magnetic field and the divergence-free condition might hamper searches. $\endgroup$ – Kyle Kanos Dec 29 '19 at 12:38
  • $\begingroup$ Isn't this the same question as "can you get shockwave solutions?" Since normal fluids do, you automatically have them in MHD unless there was some EM interaction smoothing them out. But given the number of papers on plasma shock waves I suspect there are plenty of (approximate, since below a certain length scale it is just bouncing particles) singularities. $\endgroup$ – Anders Sandberg Dec 29 '19 at 16:43
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Do the MHD equations break down under certain circumstances?

Yes, all the time. MHD is just a fluid approximation for a plasma, as I mention at https://physics.stackexchange.com/a/445307/59023 and https://physics.stackexchange.com/a/350655/59023 and https://physics.stackexchange.com/a/239769/59023 and https://physics.stackexchange.com/a/136592/59023.

There are multiple versions of MHD:

  • Ideal: infinitely conductive, magnetized fluid, and Ohm's law is just $\mathbf{E} = -\mathbf{v} \times \mathbf{B}$
  • Hall: allows for $\mathbf{j} \times \mathbf{B}$ term in Ohm's law
  • Two-fluid: treats the fluid as two separate populations and adds pressure gradient terms to Ohm's law
  • several others that are just adding extra terms to Ohm's law or including extra corrections.

None of these are exact by any means, so of course trying to model reality will break down. The question is by how much and whether the break down even matters. MHD is incredibly good at modeling things that are intrinsically boring, e.g., homogenous and uniform, stationary magnetized fluids. It is annoying good at modeling things that are not boring as well. I say annoying, because it does much better than one would think it should. However, it fails completely with collisionless shocks.

You may notice that the above link goes to an MHD article about magnetized shocks and discontinuities. MHD is fine to describe the asymptotic state of these things, but it misses several critical features and cannot explain how the shock actually initiates or dissipates energy. MHD is like a black box and if you provide the "correct" inputs and outputs, it can do some things quite well. Also note that ideal MHD is scaleless. That is, there are no relevant spatial or time scales. In the higher order versions of MHD one can calculate these scales, but they are not physically meaningful in the impact/effect that they actually have on plasmas.

Thus, a collisionless shock, for instance, is known to have a ramp thickness ranging from a few electron inertial lengths, $c/\omega_{pe}$, where $\omega_{ps}$ is the plasma frequency of species $s$ and $c$ is the speed of light in vacuum, to an ion inertial length, $c/\omega_{pi}$. MHD cannot handle this or even acknowledge it. The Rankine-Hugoniot conservation relations, for instance, assume the ramp is some thin region (depending on the book you read, some even say it's infinitely thin though I find this inappropriate), where thin is defined as a scale small enough such that normal particle dynamics cannot occur. That is, the change occurs faster than the particles can respond. As noted above, when the ramp is smaller than the ion scales, these particles cannot undergo a complete gyro orbit within the ramp.

In simplistic terms, could two (high density) rings of plasma colliding lead to the emergence of regions of high density (temperature, pressure) plasma, that could facilitate fusion in a tokamak (or stellarator), for example?

I am not sure I follow, but coronal mass ejections often overtake previously launched ones forming shock-shock interaction regions.

As an aside, although high energies are necessary to get particles past their own Coulomb potentials, high energies also make the plasma more likely to escape. That is, in the case of fusion experiments, the biggest obstacle is and has always been confinement. Confinement just refers to keeping the plasma held in place long enough for enough particles to undergo fusion that the energy output meets or exceeds the energy input used to generate these conditions, also called the break even point.

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  • $\begingroup$ I will try to study the reference links recommended . I am not a physicist, I am a mathematician. I slso realise the necessary familiarity with the work of Euler, Maxwell, and Chandrasekhar in this context. I doubt that "massaging" plasmas with smooth magnetic fields will ever lead to (regardless of the geometry) successful fusion experiments. Shockwave dynamics might offer some hope. Thank you @honeste_vivere for your answer. $\endgroup$ – Cristian Dumitrescu Dec 29 '19 at 23:34
  • $\begingroup$ My guess (intuition based only) is that if the "colliding rings" configurations are interesting in the context of Euler equations, maybe something similar would be interesting to study in MHD. If these rings are sufficiently small and the collisions take place at the center of a large stellarator, you probably don't have to worry a lot about confinement, and yet have enough fusion processes taking place in certain regions around the center. $\endgroup$ – Cristian Dumitrescu Dec 29 '19 at 23:40
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    $\begingroup$ Those colliding plasma rings sounds a lot like some work done at a fusion startup called TAE Technologies en.wikipedia.org/wiki/TAE_Technologies. $\endgroup$ – Maxim Umansky Dec 30 '19 at 4:53
  • $\begingroup$ Let's get into some heuristics (amateur hour now, please excuse ). Let's consider two such rings of plasma waves colliding (same diameter, that's essential). The velocity of the ring (waves ) is not sufficient to overcome the Coulomb barrier in direct collision. When these rings collide, let's consider a 50% change that the particles (ions, electrons, whatever) will bounce outwards or inwards (towards the interior of the rings, 50% seems like a reasonable assumption, see the link in my question). $\endgroup$ – Cristian Dumitrescu Dec 30 '19 at 8:24
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    $\begingroup$ CristianDumitrescu - Are you asking about plasmas produced in systems like the SSX experiment at Swarthmore (swarthmore.edu/ssx-lab)? It's a coaxial tube that launches a toroidal plasma into chamber bounded by Helmholtz coils. I think there are similar to experiments but not one-sided like SSX. That is, they can launch two of these toroidal plasmas at each other to force interaction. Other lab experiments launch CME-like blobs at each other to initiate shocks and study other phenomena. $\endgroup$ – honeste_vivere Dec 30 '19 at 16:04

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