Talk-Polywell.org

The version of Maxwell's equations commonly in use assumes that magnetic monopoles cannot exist. Certainly we don't observe them. If they did, though, the maths would still be tractable.

As plasma instability is proving a major challenge, is it conceivable that magnetic monopoles could fleetingly exist in a plasma, rendering the assumptions of the model invalid?

Not likely, a plasma isn't that esoteric physically. We almost certainly would have seen monopoles at particle accelerator energies already if that were possible.

Hydrodynamic stability is still a major challenge, i.e. fluid dynamic turbulence, still one of the great unsolved puzzles of physics last time I looked. Navier-Stokes equations have yet to yield a closed form solution (hence my skepticism of all GFD GCM models).

It is no surprise that magneto-hydrodynamic stability is a "challenge".

Plasma solutions are basically Navier-Stokes fueled on Maxwell equation steroids. While the Maxwell equations maybe linear and tractable, the Navier-Stokes are not, it is that simple I think.

Like this, perhaps:

http://www.sciencemag.org/cgi/content/abstract/1167747

http://en.wikipedia.org/wiki/Magnetic_monopole

Elsewhere, I found a comment that magnetic monopoles must not exceed 1 in 10^29 nucleons else we'd have noticed their effects...

A perhaps whimsical comment argued, "Sure, they exist-- But the first Galactic-centre black holes swept them up"...

icarus wrote:While the Maxwell equations maybe linear and tractable, the Navier-Stokes are not, it is that simple I think.

A million dollar prize awaits here:
http://www.claymath.org/millennium/index.php

But a couple of interesting ideas here, though:

Quaternions and particle dynamics in the Euler fluid equations
http://www.ma.ic.ac.uk/~jdg/nonlinquat.pdf

Intermittency and Regularity Issues in 3D Navier-Stokes Turbulence
http://www.ma.ic.ac.uk/~jdg/armajour.pdf

Perhaps turbulence is a chaotic system, in the mathematical sense of "chaos." If so, it might be possible to "control" a turbulent system by finding an attractor which is beneficial, i.e. a situation where flow is smooth, and then figuring out how to supply inputs to keep the system in that attractor. I attended a lecture at Sandia Labs where this technique was applied to machine tool chatter. The researcher had done work where, rather than looking just at the machine system resonances, he had treated the chatter as a chaotic system, and found a way to keep a machine tool operating in a low-chattering attractor by carefully applying timed forces to keep it in the attractor. It was a neat solution and it seemed like the principal could apply to any chaotic system--find an attractor that you like and work to keep the system there. I'm not sure how one would apply input into a turbulent plasma system to try to keep it in a usefully smooth-moving attractor, but it's possible that it could be done.

icarus wrote:Navier-Stokes equations have yet to yield a closed form solution.

Not true. Laminar solutions are easy. Hagen-Poiseuille flow, for example (fully-developed laminar pipe flow) is a very simple exact solution to the Navier-Stokes equations. I've had to work it out myself on a couple of occasions because my textbooks were somewhere else; it's a back-of-the-envelope job. Of course, it's only physically realistic for low Reynolds numbers (the usual engineering approximation is Re_D < 2300), and turbulent flow has to be modelled empirically or numerically. The transition between laminar and turbulent is particularly sticky, since it depends so strongly on slight irregularities resulting in the growth of instabilities; even the mighty Moody diagram can't give you much useful information in that regime...

Plasma solutions are basically Navier-Stokes fueled on Maxwell equation steroids.

Not really true either. For many plasmas of interest in aerospace engineering (and presumably in Polywell), Navier-Stokes and the underlying assumption of local thermodynamic equilibrium (well, almost, or there would be no viscosity or thermal conductivity) are inadequate. For systems further out of equilibrium, moment methods are required. For systems very far out of equilibrium, only Boltzmann's equation will do. The collision operator in Boltzmann's equation accounts for everything from the Maxwellianization process itself to viscosity and thermal conductivity, and is almost totally intractable without simplifying assumptions (such as Fokker-Planck, which IIRC is specifically for inverse-square interparticle potentials and is thus mostly useful for charged particles).

On topic, I don't see what in a plasma would result in a magnetic monopole. It's essentially a current field, so ye olde Biot-Savart law should give you the magnetic field, and since the current is an ordinary charge-conserving electrodynamic phenomenon, the magnetic field should be fairly ordinary too.

93143:

Thanks for clarifying my short statements, english generalisations of mathematical statements are often inadequate.

Of course, I meant,

"Navier-Stokes equations have yet to yield a general closed form solution."

I think at last count there are 13 or 14 known exact solutions, most of them are in the text of Schlicting's "Boundary-Layer Theory". Only one since then involving flow inside a cylindrical container with a rotating lid. They are all for specific simplified geometries and boundary/initial conditions.

Maybe Navier-Stokes has nothing to do with why plasma's are a challenge but if the simplified (equilibrium) cases are intractable, it is hard to see why the non-equilibrium cases would be less of a challenge. I don't buy that, simply my opinion.

icarus wrote:Thanks for clarifying my short statements, english generalisations of mathematical statements are often inadequate.

You're welcome. I wonder if I've just lost some of the moral high ground re: criticizing chrismb for nitpicking...

Maybe Navier-Stokes has nothing to do with why plasma's are a challenge but if the simplified (equilibrium) cases are intractable, it is hard to see why the non-equilibrium cases would be less of a challenge. I don't buy that, simply my opinion.

Ever heard of the 20-moment model? That's right - 20 equations in 20 variables (compared to 5 for Euler and Navier-Stokes). And it's still not general enough to tackle a Polywell. I didn't say it was less of a challenge... BTW Boltzmann's equation is a six-dimensional equation, not counting time; now imagine trying to do DNS of a turbulent flow with sufficient resolution in all six dimensions...

In other news, it turns out I was wrong about the Fokker-Planck approximation. It's useful for any interparticle potential for which most interactions result in small changes to the particle paths. So, generally low-order, like inverse-square, but not specifically limited to inverse-square...

This thread is relevant to my interests.

Seriously, that adds a fair amount to my layman's understanding of the basic concepts involved. Thanks for sharing.